Antenna processing method for potentially non-centered cyclostationary signals

ABSTRACT

An antenna processing method for centered or potentially non-centered cyclostationary signals, comprises at least one step in which one or more nth order estimators are obtained from r-order statistics, with r=1 to n−1, and for one or more values of r, it comprises a step for the correction of the estimator by means of r-order detected cyclic frequencies. The method can be applied to the separation of the emitter sources of the signals received by using the estimator or estimators.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The invention relates to an antenna processing method forpotentially non-centered or centered cyclostationary sources.

[0003] It can be applied for example to CPFSK sources with integermodulation index.

[0004] The invention relates, for example, to a method for theseparation of potentially non-centered, cyclostationary signals receivedby a receiver of a communications system comprising several sources oremitters. The term “cyclostationary” also designates the particular caseof stationary signals.

[0005] It can also be applied to the angular localization or goniometryof potentially non-centered cyclostationary sources.

[0006] The invention can be applied especially in radiocommunications,space telecommunications or passive listening to these links, infrequencies ranging from the VLF (Very Low Frequency) to the EHF(Extremely High Frequency).

[0007] In the present description, the term “blind separation”designates the separation of emitters with no knowledge whatsoever ofthe signals sent, the term “centered signal” refers to a signal withoutany continuous component that verifies E[x(t)]=0, and the term“non-stationary signal” refers to a signal whose statistics aretime-dependent.

[0008] 2. Description of the Prior Art

[0009] In many contexts of application, the reception of signals ofinterest for the receiver is very often disturbed by the presence ofother signals (or sources) known as parasites, which may correspondeither to delayed versions of the signals of interest (throughmultiple-path propagation), or to interfering sources which may beeither deliberate or involuntary (in the case of co-channeltransmissions). This is especially the case with radiocommunications inurban areas, subject to the phenomenon of multiple paths resulting fromthe reflections of the signal on surrounding fixed or moving obstaclespotentially disturbed by the co-channel transmissions coming from theneighboring cells that re-use the same frequencies (in the case ofF/TDMA or Frequency/Time Division Multiple Access networks). This isalso the case with the HF (High Frequency) ionospherical links disturbedby the presence, at reception, of the multiple paths of propagationresulting from the reflections on the different ionospherical layers andof parasitic emitters due to high spectral congestion in the HF range.

[0010] For all these applications, whether it is for purposes ofradiocommunications or for listening and technical analysis of thesources received, the sources need to be separated before otherprocessing operations specific to the application considered areimplemented. Furthermore, for certain applications such as passivelistening, the sources received are totally unknown to the receiver(there are no available learning sequences, the waveforms are unknown,etc.) and their angular localization or goniometry may prove to bedifficult (because of coupling between sensors) or costly (because ofthe calibration of the aerials) to implement. This is why it may proveto be highly advantageous to implement a source separation technique ina totally autodidactic or self-learning way, that is, by making use ofno a priori information on the sources, apart from the assumption of thestatistical independence of these sources.

[0011] The first studies on the separation of sources by self-learningappeared in the mid-1980s in the work of Jutten and Herault [1]. Sincethen, these studies have been constantly developing for mixtures ofsources, both convolutive (time-spread multiple-path propagationchannels) and instantaneous (distortion-free channels). A conspectus ofthese studies is presented in the article [2] by P. Comon and P.Chevalier. A certain number of techniques developed are calledsecond-order techniques because they use only the information containedin the second-order statistics of the observations, as described inreference [3] for example. By contrast, other techniques, known ashigher-order techniques, described for example in the reference [4],generally use not only second-order information but also informationcontained in statistics above the second order. These include thetechniques known as cumulant-based, fourth-order techniques which havereceived special attention owing to their performance potential(reference [2]) and the relative simplicity of their implementation.

[0012] However, almost all the techniques of self-learned sourceseparation available to date have been designed to separate sourcesassumed to be stationary, centered and ergodic, on the basis ofestimators of statistics of observations qualified as being empirical,asymptotically unbiased and consistent on the basis of the aboveassumptions.

[0013] Two families of second-order separators are presently available.Those of the first family (F1) (reference [3], using the SOBI methodshown schematically in FIG. 1) are aimed at separating statisticallyindependent sources assumed to be stationary, centered and ergodicwhereas those of the second family (F2) (reference [6], using the cyclicSOBI method) are designed to separate statistically independent sourcesassumed to be cyclostationary, centered and cycloergodic.

[0014] Two families of fourth-order separators are presently available.For example, those of the first family (F3) (reference [4] by J. F.Cardoso and A. Souloumiac, using the JADE method shown schematically inFIG. 2) are aimed at separating statistically independent sourcesassumed to be stationary, centered and ergodic while those of the secondfamily (F4) (reference A. Ferreol and Chevalier [8] using the cyclicJADE technique) are designed to separate statistically independentsources assumed to be cyclostationary, centered and cycloergodic.

[0015] However, most of the sources encountered in practice arenon-stationary and, more particularly, cyclostationary (with digitallymodulated sources) and in certain cases deterministic (pure carriers).Furthermore, some of these sources are not centered. This is especiallythe case for deterministic sources and for certain digitally andnon-linearly modulated sources as in the case of CPFSK sources withinteger modulation index. This means that the empirical estimators ofstatistics classically used to implement the current techniques ofself-learned source separation no longer have any reason to remainunbiased and consistent but are liable to become asymptotically biased.This may prevent the separation of the sources as shown in the documentby A. Ferreol and P. Chevalier [5] for centered cyclostationary sources(linear digital modulations).

SUMMARY OF THE INVENTION

[0016] The invention relates to an antenna processing method forcentered or potentially non-centered cyclostationary signals, comprisingat least one step in which an nth order statistics estimator is obtainedfrom r-order statistics, with r=1 to n−1, and for one or more values ofr, a step for the correction of the empirical estimators, by means ofr-order detected cyclic frequencies, exploiting the potentiallynon-centered character of the observations.

[0017] It comprises for example a step for the separation of the emittersources of the signals received by using one of the second-orderestimators or fourth-order estimators proposed.

[0018] It is also used for the angular localization or goniometry of thesignals received.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Other features and advantages of the inventions shall appear moreclearly from the following description of a non-exhaustive exemplaryembodiment and from the appended figures, of which:

[0020]FIGS. 1 and 2 show prior art separation techniques, respectivelyknown as the SOBI and JADE techniques,

[0021]FIG. 3 exemplifies a receiver according to the invention,

[0022]FIG. 4 is a diagram of the steps according to the inventionapplied to the SOBI technique of separation,

[0023]FIG. 5 is a diagram of the steps of the method according to theinvention applied to the JADE technique of separation,

[0024]FIGS. 6A and 6B show results of separation using a classicestimator or an estimator according to the invention,

[0025]FIG. 7 shows the convergence of the estimator,

[0026]FIG. 8 is a drawing of the steps implemented according to theinvention for the cyclic JADE method.

MORE DETAILED DESCRIPTION

[0027] To put it briefly, the method according to the invention enablesthe processing especially of potentially non-centered or centeredcyclostationary sources. It can be applied, for example, to theseparation of the sources or to their angular localization orgoniometry. It comprises at least one step using an nth order statisticsestimator constructed especially in taking account of the r-orderstatistics, with r varying from 1 to n−1, and for one or more values ofr, a step for the correction of the estimator from the detected r-ordercyclic frequencies, making use of the potentially non-centered characterof the observations.

[0028]FIG. 1 gives a diagrammatic view of an exemplary receivercomprising, for example, N reception sensors (1 ₁, 1 ₂, 1 ₃, 1 ₄) eachconnected to several inputs of a processing device 2 such as a processoradapted to executing the steps of the method described here below.

[0029] Before describing various alternative modes of implementation ofthe method according to the invention, we shall describe the way toobtain the estimators, for example second-order or fourth-orderestimators, as well as the detection of the cyclic frequencies.

[0030] Modelling of the Signal

[0031] It is assumed that an antenna with N sensors receives anoise-ridden mixture from P (P≦N) narrow-band (NB) and statisticallyindependent sources. On the basis of these assumptions, the vector, x(t)of the complex envelopes of the output signals from the sensors iswritten, at the instant t, as follows: $\begin{matrix}{{x(t)} = {{{\sum\limits_{p = 1}^{P}\quad {{m_{p}(t)}^{j{({{2\pi \quad \Delta \quad f_{p}t} + \varphi_{p}})}}a_{p}}} + {b(t)}}\overset{\Delta}{=}{{A\quad {m_{c}(t)}} + {b(t)}}}} & (1)\end{matrix}$

[0032] where b(t) is the noise vector, assumed to be centered,stationary, circular and spatially white, m_(p)(t), Δf_(p), φ_(p) anda_(p) correspond respectively to the complex, narrow-band (NB),cyclostationary and potentially non-centered envelope (deterministic asthe case may be), to the residue of the carrier, to the phase and to thedirectional vector of the source p, m_(c)(t) is the vector whosecomponents are the signals m_(pc)(t) ^(Δ) m_(p)(t)exp[j(2πΔf_(p)t+φ_(p))] and A is the matrix (N×P) whose columns are thevectors a_(p).

[0033] Statistics of the Observations

[0034] First Order Statistics

[0035] In the general case of cyclostationary and non-centered sources,the first-order statistics of the vector x(t), given by (1), are writtenas follows: $\begin{matrix}{{e_{x}(t)}\overset{\Delta}{=}{{E\lbrack {x(t)} \rbrack} = {{\sum\limits_{p = 1}^{P}{{e_{p}(t)}^{j{({{2\pi \quad \Delta \quad f_{p}t} + \varphi_{p}})}}a_{p}}}\overset{\Delta}{=}{{\sum\limits_{p = 1}^{P}{{e_{pc}(t)}a_{p}}}\overset{\Delta}{=}{A\quad {e_{m\quad c}(t)}}}}}} & (2)\end{matrix}$

[0036] where e_(p)(t), e_(pc)(t) and e_(mc)(t) are the mathematicalexpectation values respectively of m_(p)(t), m_(pc)(t) and m_(c)(t). Thevectors e_(p)(t) and e_(pc)(t) accept a Fourier series decomposition,and we obtain: $\begin{matrix}{{e_{pc}(t)}\overset{\Delta}{=}{{E\lbrack {m_{pc}(t)} \rbrack} = {{\sum\limits_{\gamma_{pc} \in \Gamma_{pc}^{1}}{e_{pc}^{\gamma_{pc}}^{{j2\pi\gamma}_{pc}t}}} = {\sum\limits_{\gamma_{p} \in \Gamma_{p}^{1}}{e_{p}^{\gamma_{p}}^{j{\lbrack{{2{\pi(\quad {{\Delta \quad f_{p}} + \gamma_{p}})}t} + \varphi_{p}}\rbrack}}}}}}} & (3)\end{matrix}$

[0037] whereΓ_(p)¹ = {γ_(p)}  and  Γ_(pc)¹ = {γ_(pc) = γ_(p) + Δ  f_(p)}  

[0038] are the sets of the cyclic frequencies γ_(p) and γ_(pc)respectively of e_(p)(t) and e_(pc)(t),e_(p)^(γ_(p))  and  e_(pc)^(γ_(pc))

[0039] are the cyclic mean values respectively m_(p)(t) and m_(pc)(t),defined by $\begin{matrix}{e_{p}^{\gamma_{p}}\overset{\Delta}{=}{< {{e_{p}(t)}^{{- {j2\pi\gamma}_{p}}t}} >_{c}}} & (4) \\{e_{pc}^{\gamma_{pc}}\overset{\Delta}{=}{{< {{e_{pc}(t)}^{{- {j2\pi\gamma}_{pc}}t}} >_{c}} = {e_{p}^{\gamma_{{pc} - {\Delta \quad {fp}}}}^{{j\varphi}_{p}}}}} & (5)\end{matrix}$

[0040] where the symbol${< {f(t)} >_{c}}\overset{\Delta}{=}\quad {\lim\limits_{Tarrow\infty}{( {1/T} ){\int_{{- T}/2}^{T/2}{{f(t)}\quad {t}}}}}$

[0041] corresponds to the operation of taking the temporal mean incontinuous time f(t) on an infinite horizon of observation.Consequently, the vectors e_(mc)(t) and e_(x)(t) also accept a Fourierseries decomposition and, by using (2) and (3), we get: $\begin{matrix}{{e_{m\quad c}(t)}\overset{\Delta}{=}{\sum\limits_{\gamma \in \Gamma_{x}^{1}}\quad {e_{m\quad c}^{\gamma}^{{j2\pi\gamma}\quad t}}}} & (6) \\{{e_{x}(t)}\overset{\Delta}{=}{{\sum\limits_{\gamma \in \Gamma_{x}^{1}}{e_{x}^{\gamma}^{{j2\pi\gamma}\quad t}}} = {{\sum\limits_{\gamma \in \Gamma_{x}^{1}}{A\quad e_{m\quad c}^{\gamma}^{{j2\pi\gamma}\quad t}}} = {\sum\limits_{p = 1}^{P}{\sum\limits_{\gamma_{pc} \in \Gamma_{pc}^{1}}{e_{pc}^{\gamma_{pc}}^{{j2\pi\gamma}_{pc}t}a_{p}}}}}}} & (7)\end{matrix}$

[0042] where Γ_(x)¹ = ⋃_(1 ≤ p ≤ p){Γ_(pc)¹}

[0043] is the set of the cyclic frequencies γ of e_(mc)(t) and e_(x)(t),e_(m  c)^(γ)

[0044] and e_(x)^(γ)  and  e_(x)^(γ)

[0045] are respectively the cyclic means of m_(c)(t) and x(t), definedby: $\begin{matrix}{e_{m\quad c}^{\gamma} = {< {{e_{m\quad c}(t)}^{- {j2\pi\gamma t}}} >_{c}}} & (8) \\{e_{x}^{\gamma} = {< {{e_{x}(t)}^{- {j2\pi\gamma t}}} >_{c}}} & (9)\end{matrix}$

[0046] Assuming these conditions, the (quasi)-cyclostationary vectorx(t) can be decomposed into the sum of a deterministic and(quasi)-periodic part e_(x)(t) and a (quasi)-cyclostationary, centered,random part Δx(t) such that:

Δx(t) ^(Δ) x(t)−e _(x)(t)=AΔm _(c)(t)+b(t)  (10)

[0047] where Δm_(c)(t) ^(Δ) m_(c)(t)−e_(mc)(t) is the centered vector ofthe source signals, with componentsΔm_(pc)(t)Δm_(pc)(t)−e_(pc)(t)=Δm_(p)(t) e^(j(2πΔf) ^(_(p)) ^(t+φ)^(_(p)) ⁾where Δm_(p)(t) ^(Δ) m_(p)(t)−e_(p)(t).

[0048] Special Cases

[0049] By way of an indication, e_(p)(t)=0 for a digitally and linearlymodulated source p, which is centered. However, e_(p)(t)≠0 for adeterministic (carrier) source p as well as for certain digitally andnon-linearly modulated sources such as CPFSKs with integer index, whosecomplex envelope is written as follows: $\begin{matrix}{{m_{p}(t)} = {\pi_{p}^{1/2}{\sum\limits_{n}\quad {\exp \{ {j\lbrack {\theta_{pn} + {2\pi \quad f_{dp}{a_{n}^{p}( {t - {n\quad T_{p}}} )}}} \rbrack} \} \quad {{Rect}_{p}( {t - {n\quad T_{p}}} )}}}}} & (11)\end{matrix}$

[0050] where T_(p) corresponds to the symbol duration of the source,π_(p) ^(Δ) <E[|m_(p)(t)|²]>_(c) is the mean power of the source preceived by an omnidirectional sensor, the a_(n)^(p)

[0051] are the transmitted M_(p)-ary symbols, assumed to be i.i.d andtaking their values in the alphabet ±1, ±3, . . . , ±(M_(p)−1), whereM_(p) is generally a power of 2, Rect_(p)(t) is the rectangular pulsewith an amplitude 1 and a duration T_(p), f_(dp) ^(Δ) h_(p)/2T_(p) isthe frequency deviation, h_(p) is the index of modulation of the sourceand θ_(pn), which corresponds to the accumulation of all the symbols upto the instant (n−1)T_(p), is defined by: $\begin{matrix}{\theta_{pn}\overset{\Delta}{=}{2\pi \quad f_{dp}T_{p}{\sum\limits_{k = {- \infty}}^{n - 1}\quad a_{k}^{p}}}} & (12)\end{matrix}$

[0052] For M_(p)-ary symbols, the associated CPFSK source is called anM_(p)-CPFSK source. For this type of source, the set Γ_(p) ¹ of thefirst-order cyclic frequencies of the source p is written asΓ_(p)¹ = {γ_(p) = ±(2k + 1)f_(dp),

[0053] 0≦k≦(M_(p)−2)/2}. Thus we get: $\begin{matrix}{e_{p}^{\gamma} = {{{\pm \pi_{p}^{1/2}}\frac{1}{M_{p}}\quad {for}\quad \gamma} \in \Gamma_{p}^{1}}} & (13)\end{matrix}$

[0054] Second-Order Statistics

[0055] Based on the above assumptions (non-centered, cyclostationarysources), the second-order statistics of the observations arecharacterized by the two correlation matrices R_(xε)(t, τ) for ε=1 andε=−1, dependent on the current time t and defined by:

R _(xε)(t,τ) ^(Δ) E[x(t)x(t−τ)^(ε) ^(_(T)) ]=AR _(mcε)(t,τ)A ^(ε)^(_(T)) +η2(τ)δ(1+ε)I  (14)

[0056] where ε=±1, with the convention x^(1Δ) x and x^(−1Δ) x*, * is thecomplex conjugation operation, δ(.) is the Kronecker symbol, ^(T)signifies transposed, η₂(τ) is the correlation function of the noise oneach sensor, I is the identity matrix, the matrix R_(mcε)(t, τ) ^(ΔE[m)_(c)(t) m_(c)(t−τ)^(ε) ^(_(T)) ] introduces the first and secondcorrelation matrices of the vector m_(c)(t).

[0057] In the general case of non-centered cyclostationary sources using(10) in (14), the matrix R_(xε)(t, τ) takes the following form:

R _(xε)(t,τ)=R _(Δxε)(t,τ)+e _(x)(t)e _(x)(t−τ)^(ε) ^(_(T))   (15)

[0058] where R_(Δxε)(t, τ) introduces the first and second matrices ofcovariance of x(t) or of correlation of Δx(t) , defined by

R _(Δxε)(t,τ) ^(Δ) E[Δ x(t)Δx(t−τ)^(ε) ^(_(T)) ]=AR _(Δmcε)(t, τ)A ^(ε)^(_(T)) +η₂(τ)δ(1+ε)I  (16)

[0059] where R_(Δmcε)(t, τ) ^(Δ) E[Δm_(c)(t) Δm_(c)(t−τ)^(ε) ^(_(T)) ]defines the first and second matrices of covariance of m_(c)(t) suchthat:

R _(mcε)(t,τ)=R _(Δmcε)(t,τ)+e _(mc)(t)e _(mc)(t−τ)^(ε) ^(_(T))   (17)

[0060] Using (1), (10) and the assumption of statistical independence ofthe sources in (16), we get: $\begin{matrix}\begin{matrix}{{R_{\Delta \quad {xɛ}}( {t,\tau} )} = {{\sum\limits_{p = 1}^{P}\quad {{r_{\Delta \quad p\quad ɛ}( {t,\tau} )}^{j{\lbrack{{{({1 + ɛ})}{({{2{\pi\Delta}\quad f_{p}t} + \varphi_{p}})}} - {2{\pi\Delta}\quad f_{p}{\tau ɛ}}}\rbrack}}a_{p}a_{p}^{ɛ\quad T}}} + {{\eta_{2}(\tau)}I}}} \\{= {{\sum\limits_{p = 1}^{P}\quad {{r_{\Delta \quad p\quad {cɛ}}( {t,\tau} )}a_{p}a_{p}^{ɛ\quad T}}} + {{\eta_{2}(\tau)}{\delta ( {1 + ɛ} )}I}}}\end{matrix} & (18)\end{matrix}$

[0061] where r_(Δpε)(t, τ) ^(Δ) E[Δm_(p)(t) Δm_(p)(t−τ)^(ε)],r_(Δpcε)(t, τ) ^(Δ) E[Δm_(pcε)(t) Δm_(pc)(t−τ)^(ε)]. From theexpressions (2) and (18) a new writing of the expression (15) is finallydeduced and is given by: $\begin{matrix}{{R_{xɛ}( {t,\tau} )} = {{\sum\limits_{p = 1}^{P}{{r_{\Delta \quad p\quad {cɛ}}( {t,\tau} )}a_{p}a_{p}^{ɛ\quad T}}} + {\sum\limits_{p = 1}^{P}{\sum\limits_{q = 1}^{P}{{e_{pc}(t)}{e_{qc}( {t - \tau} )}^{ɛ}a_{p}a_{p}^{ɛ\quad T}}}} + {{\eta_{2}(\tau)}{\delta ( {1 + ɛ} )}I}}} & (19)\end{matrix}$

[0062] For cyclostationary sources, the functions of covariancer_(Δpcε)(t, τ), 1≦p≦P, and hence the matrices R_(Δmcε)(t, τ) andR_(Δxε)(t, τ) are (quasi-) or polyperiodic functions of the time taccepting a Fourier series decomposition, and we get: $\begin{matrix}{{r_{\Delta \quad p\quad c\quad ɛ}( {t,\tau} )} = {\sum\limits_{\alpha_{\quad^{{\Delta \quad p\quad c\quad ɛ} \in {\Gamma\Delta}_{pc}^{\lbrack{1,ɛ}\rbrack}}}}{{r_{\Delta \quad {pc}\quad ɛ}^{\alpha_{\Delta \quad p\quad c\quad ɛ}}(\tau)}^{{j2\pi}\quad \alpha_{\Delta \quad {pc}\quad ɛ\quad t}}}}} & (20) \\{{R_{\Delta \quad x\quad ɛ}( {t,\tau} )} = {\sum\limits_{\alpha_{\quad^{{\Delta \quad x\quad ɛ} \in {\Gamma\Delta}_{x}^{\lbrack{1,ɛ}\rbrack}}}}{{R_{\Delta \quad x\quad ɛ}^{\alpha_{\Delta \quad x\quad ɛ}}(\tau)}^{{j2\pi}\quad \alpha_{\Delta \quad x\quad ɛ\quad t}}}}} & (21)\end{matrix}$

[0063] where Γ_(Δ  pc)^([1, ɛ]) = {α_(Δ  pc  ɛ)}

[0064] is the set of the cyclic frequencies α_(Δpcε)ofr_(Δ  pc  ɛ)(t, τ), Γ_(Δ  x)^([1, ɛ]) = {α_(Δ  x  ɛ)} = ⋃_(1 ≤ p ≤ P){Γ_(Δ  pc)^([1, ɛ])}

[0065] is the set of cyclic frequencies α_(Δxε)of R_(Δxε)(t, τ), thequantities r_(Δ  pc  ɛ)^(α_(Δp  c  ɛ))(τ)

[0066] define the first (ε=−1) and second (ε=1) functions of cycliccovariance of m_(pc)(t) while the matricesR_(Δ  x  ɛ)^(α_(Δ  x  ɛ))(τ)

[0067] define the first and second matrices of cyclic covariance ofx(t), respectively defined by $\begin{matrix}{{r_{\Delta \quad {pc}\quad ɛ},^{\alpha_{\Delta \quad {pc}\quad ɛ}}{(\tau) = {< {{r_{\Delta \quad {pc}\quad ɛ}( {t,\tau} )}^{{- j}\quad 2\pi \quad \alpha_{\Delta \quad {pc}\quad ɛ}t}} >_{c}}}}\quad} & (22) \\{R_{\Delta \quad x\quad ɛ},^{\alpha_{\Delta \quad x\quad ɛ}}{(\tau) = {{< {{R_{\Delta \quad x\quad ɛ}( {t,\tau} )}^{{- j}\quad 2\pi \quad \alpha_{\Delta \quad x\quad ɛ}t}} >_{c}} = {{A\quad {R_{\Delta \quad m\quad c\quad ɛ}^{\alpha_{\Delta \quad x\quad ɛ}}(\tau)}A^{ɛ\quad T}} + {{\eta_{2}(\tau)}{\delta ( \alpha_{\Delta \quad x\quad ɛ} )}{\delta ( {1 + ɛ} )}I}}}}} & (23)\end{matrix}$

[0068] where R_(Δ  m  c  ɛ)^(α_(Δ  x  ɛ))(τ)

[0069] defines the first and second matrices of cyclic covariance ofm_(c)(t). Using the expressions (7) and (21) in (15), we get$\begin{matrix}{{R_{x\quad ɛ}( {t,\tau} )} = {{\sum\limits_{\alpha_{\Delta \quad x\quad ɛ} \in \quad \Gamma_{\Delta_{x}}^{\lbrack{1,ɛ}\rbrack}}{{R_{\Delta \quad x\quad ɛ}^{\alpha_{\Delta \quad x\quad ɛ}}(\tau)}^{j\quad 2\quad {\pi\alpha}_{\Delta \quad x\quad ɛ}t}}} + {\sum\limits_{\gamma \quad \in \quad \Gamma_{x}^{1}}{\sum\limits_{\omega \quad \in \quad \Gamma_{x}^{1}}{e_{x}^{\gamma}e_{x}^{\omega \quad ɛ\quad T}^{{- j}\quad 2\pi \quad \omega \quad \tau \quad ɛ}^{j\quad 2{\pi {({\gamma + {\omega \quad ɛ}})}}t}}}}}} & (24)\end{matrix}$

[0070] which shows that the matrices R_(xε)(t, τ) accept a Fourierseries decomposition $\begin{matrix}{{R_{x\quad ɛ}( {t,\tau} )} = {\sum\limits_{\alpha_{x\quad ɛ} \in \quad \Gamma_{\quad_{x}}^{\lbrack{1,ɛ}\rbrack}}{{R_{x\quad ɛ}^{\alpha_{x\quad ɛ}}(\tau)}^{j\quad 2\quad {\pi\alpha}_{x\quad ɛ}t}}}} & (25)\end{matrix}$

[0071] whereΓ_(x)^([1, ɛ]) = {α_(x  ɛ)} = Γ_(Δ  x)^([1, ɛ])⋃{Γ_(x)¹o_(ɛ)Γ_(x)¹}

[0072] is the set of the cyclic frequencies α_(xε) ofR_(x  ɛ)(t, τ), Γ_(x)¹o_(ɛ)Γ_(x)¹ = {α = γ + ɛω  where  γ ∈ Γ_(x)¹, ω ∈ Γ_(x)¹},

[0073] the matrices R_(x  ɛ)^(α_(x  ɛ))(τ)

[0074] are the first (ε=−1) and second (ε=+1) matrices of cycliccorrelation of x(t), defined by $\begin{matrix}{{R_{x\quad ɛ}^{\alpha_{x\quad ɛ}}(\tau)} = {{< {{R_{x\quad ɛ}( {t,\tau} )}^{{- j}\quad 2\pi \quad \alpha_{x\quad ɛ}t}} >_{c}} = {{A\quad {R_{m\quad c\quad ɛ}^{\alpha_{x\quad ɛ}}(\tau)}A^{ɛ\quad T}} + {{\eta_{2}(\tau)}\quad {\delta ( \alpha_{x\quad ɛ} )}\quad {\delta ( {1 + ɛ} )}\quad I}}}} & (26)\end{matrix}$

[0075] where R_(mc  ɛ)^(α_(x  ɛ))(τ)

[0076] defines the first and second matrices of cyclic correlation ofm_(c)(t). In particular, for the zero cyclic frequency, the matrixR_(x  ɛ)^(α_(x  ɛ))(τ)

[0077] corresponds to the temporal mean in t, R_(xε)(τ), de R_(xε)(t, τ)which is written, using (14),

R _(xε)(τ) ^(Δ) <R _(xε)(t,τ)>_(c) =AR _(mcε)(τ)A^(εT)+η2(τ)δ(1+ε)I  (27)

[0078] where, using (17), the matrix R_(mcε)(τ) ^(Δ) <R_(mcε)(t, τ)>_(c)is written

R _(mcε)(τ) ^(Δ) <R _(mcε)(t,τ)>_(c) =R _(Δmcε)(τ)+<e _(mc)(t)e_(mc)(t−τ)^(εT)>_(c) =R _(Δmcε)(τ)+E _(mcε)(τ)  (28)

[0079] where R_(Δmcε)(τ) ^(Δ) <R_(Δmcε)(t, τ)>_(c) and E_(mcε)(τ) ^(Δ)<e_(mc)(t) e_(mc)(t−τ)^(εT)>c.

[0080] Empirical Estimator of the Second-Order Statistics

[0081] In practice, the second-order statistics of the observations areunknown in principle and must be estimated by the taking the temporalmean on a finite period of observation, on the basis of a number K ofsamples x(k) (1≦k≦K), of the observation vector x(t), using the propertyof ergodicity of these samples in the stationary case and ofcycloergodicity of these samples in the cyclostationary case. If T_(e)denotes the sampling period, the estimation of the cyclic correlationmatrix R_(x  ɛ)^(α)(τ)

[0082] for τ=lT_(e), which corresponds to a matrix of cumulants only forcentered observations, is done only by means of the estimatorR̂_(x  ɛ)^(α)(lT_(e))(K)

[0083] qualified as empirical and defined by $\begin{matrix}{{{\hat{R}}_{x\quad ɛ}^{\alpha}( {lT}_{e} )}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x(k)}\quad {x( {k - l} )}^{ɛ\quad T}^{{- j}\quad 2\pi \quad \alpha \quad {kT}_{e}}}}} & (29)\end{matrix}$

[0084] The method according to the invention comprises, for example, anovel step to determine a second-order estimator.

[0085] Novel Estimator of the Second-Order Cyclic Cumulants of theObservations

[0086] In the presence of non-centered observations, the matrices ofcorrelation of the observations R_(xε)(t, τ) no longer correspond to thematrices of covariance or of second-order cumulants of the observations,as indicated by the expression (15). This is also the case with thematrices of cyclic correlation, R_(x  ɛ)^(α_(x  ɛ))(τ),

[0087] defined by (26) which, for non-centered observations, no longercorrespond to the matrices of covariance or matrices of second-ordercyclic cumulants R_(Δ  x  ɛ)^(α_(x  ɛ))(τ),

[0088] defined by (22) for α_(Δxε)=α_(xε). Thus, an efficient operationof the second-order separators F1 and F2 with respect to the potentiallynon-centered cyclostationary sources can be obtained only by making useof the information contained in the matricesR_(Δ  x  ɛ)^(α_(x  ɛ))(τ)

[0089] rather than in the matrices R_(x  ɛ)^(α_(x  ɛ))(τ).

[0090] Thus, in as much as, for cyclostationary and cycloergodicsources, the empirical estimator (29) is an asymptotically unbiased andconsistent estimator of the matrix of cyclic correlationR_(x  ɛ)^(α)(lT_(e)),

[0091] another estimator is used for non-centered observations. Thisother estimator is aimed at making an asymptotically unbiased andconsistent estimation of the matrix of cyclic covarianceR_(Δ  x  ɛ)^(α)(lT_(e)).

[0092] From the expressions (24) and (25), we deduce the expression ofthe matrix of cyclic covariance R_(Δ  x  ɛ)^(α)(lT_(e))

[0093] as a function of that of R_(x  ɛ)^(α)(lT_(e)),

[0094] given by $\begin{matrix}{{R_{\Delta \quad x\quad ɛ}^{\alpha}( {lT}_{e} )} = {{R_{x\quad ɛ}^{\alpha}( {lT}_{e} )} - {\sum\limits_{\omega \in \Gamma_{x}^{1}}{e_{x}^{\alpha - {\omega \quad ɛ}}e_{x}^{\omega \quad ɛ\quad T}^{{- j}\quad 2\pi \quad \omega \quad ɛ\quad {lT}_{e}}}}}} & (30)\end{matrix}$

[0095] Thus, the estimation of the matrix of cyclic covarianceR_(Δ  x  ɛ)^(α)(lT_(e))

[0096] is made from the estimator R̂_(Δ  x  ɛ)^(α)(lT_(e))(K)

[0097] defined by $\begin{matrix}{{{{\hat{R}}_{\Delta \quad x\quad ɛ}^{\alpha}( {lT}_{e} )}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\quad {{\hat{R}}_{x\quad ɛ}^{\alpha}( {lT}_{e} )}(K)} - {\sum\limits_{\omega \in \Gamma_{x}^{1}}{{{\hat{e}}_{x}^{\alpha - {\omega \quad ɛ}}(K)}{{\hat{e}}_{x}^{\omega \quad}(K)}^{ɛ\quad T}^{{- j}\quad 2\pi \quad \omega \quad ɛ\quad {lT}_{e}}}}} & (31)\end{matrix}$

[0098] where R̂_(Δ  x  ɛ)^(α)(lT_(e))(K)

[0099] is defined by (29) and where ê_(x)^(ω)(K)

[0100] is defined by $\begin{matrix}{{{\hat{e}}_{x}^{\omega}(K)}\overset{\Delta}{=}{\frac{1}{K}{\sum\limits_{k = 1}^{K}\quad {{x(k)}^{{- j}\quad 2\quad \pi \quad \omega \quad {kT}_{e}}}}}} & (32)\end{matrix}$

[0101] Thus, assuming cyclostationary and cycloergodic observations,whether centered or not, the estimator (31) is an asymptoticallyunbiased and consistent estimator of the matrix of cyclic covariance ormatrix of second-order cyclic cumulantsR_(Δ  x  ɛ)^(α)(l  T_(e)).

[0102] In particular, the separators F1 must exploit the estimator (31)for α=0 and ε=−1, written as R̂_(Δ  x)(l  T_(e))(K),

[0103] and defined by $\begin{matrix}{{{{\hat{R}}_{\Delta \quad x}( {l\quad T_{e}} )}(K)}\overset{\Delta}{=}{{{{\hat{R}}_{\quad x}( {l\quad T_{e}} )}(K)} - {\sum\limits_{\omega \in \Gamma_{x}^{1}}^{\quad}\quad {{{\hat{e}}_{x}^{\omega}(K)}{{\hat{e}}_{x}^{\omega}(K)}^{\dagger}^{j\quad 2\quad \pi \quad \omega \quad l\quad T_{e}}}}}} & (33)\end{matrix}$

[0104] where {circumflex over (R)}_(x)(lT_(e))(K) is defined by (29)with α=0 and ε=−1.

[0105] According to another alternative embodiment, the method includesa step for determining a new fourth-order estimator.

[0106] Third-Order Statistics

[0107] Based on the above assumptions, the third-order statistics(moments) of the observations are defined by (34)${{T_{xɛ}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}\overset{\Delta}{=}{{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{ɛ}{x_{k}( {t - \tau_{2}} )}} \rbrack} = {{E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{j}( {t - \tau_{1}} )}^{ɛ}\Delta \quad {x_{k}( {t - \tau_{2}} )}} \rbrack} + {{e_{x}\lbrack i\rbrack}(t){E\lbrack {\Delta \quad {x_{j}( {t - \tau_{1}} )}^{ɛ}\Delta \quad {x_{k}( {t - \tau_{2}} )}} \rbrack}} + {{e_{x}\lbrack k\rbrack}( {t - \tau_{2}} ){E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{j}( {t - \tau_{1}} )}^{ɛ}} \rbrack}} + {{e_{x}\lbrack j\rbrack}( {t - \tau_{1}} )^{ɛ}{E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{k}( {t - \tau_{2}} )}} \rbrack}} + {4{e_{x}\lbrack i\rbrack}(t){e_{x}\lbrack j\rbrack}( {t - \tau_{1}} )^{ɛ}{e_{x}\lbrack k\rbrack}( {t - \tau_{2}} )}}}$

[0108] where ex_(x)[i](t)=e_(xi)(t) is the component i of e_(x)(t). Thethird-order cumulants are the quantitiesE[Δx_(i)(t)Δx_(j)(t−τ₁)^(ε)Δx_(k)(t−τ₂)]. Assuming that the termT_(xε)(t, τ₁, τ₂)[i, j, k] is the element [i, N(j−1)+k] of the matrixT_(xε)(t, τ₁, τ₂), with a dimension (N×N²), we get an expression of thismatrix given by: $\begin{matrix}{{T_{xɛ}( {t,\tau_{1},\tau_{2}} )} = {{A\quad {T_{mce}( {t,\tau_{1},\tau_{2}} )}( {A \otimes A^{ɛ}} )^{ɛ\quad T}} = {\sum\limits_{i,j,{k = 1}}^{P}\quad {{{T_{mce}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}{a_{i}\lbrack {a_{j} \otimes a_{k}^{ɛ}} \rbrack}^{ɛT}}}}} & (35)\end{matrix}$

[0109] where T_(mcε)(t, τ₁, τ₂) is the matrix (P×P²) whose coefficientsare the quantities T_(mcε)(t, τ₁, τ₂)[i, j, k] defined by$\begin{matrix}{{{T_{mce}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack} = {{E\lbrack {{m_{ic}(t)}{m_{jc}( {t - \tau_{1}} )}^{ɛ}{m_{kc}( {t - \tau_{2}} )}} \rbrack} = {{{E\lbrack {\Delta \quad {m_{ic}(t)}{m_{jc}( {t - \tau_{1}} )}^{ɛ}\Delta \quad {m_{jc}( {t - \tau_{2}} )}} \rbrack}{\delta ( {j - k} )}} + {{e_{kc}( {t - \tau_{2}} )}{E\lbrack {\Delta \quad {m_{ic}(t)}{m_{ic}( {t - \tau_{1}} )}^{ɛ}} \rbrack}{\delta ( {i - j} )}} + {{e_{jc}( {t - \tau_{1}} )}^{ɛ}{E\lbrack {\Delta \quad {m_{ic}( {t - \tau_{2}} )}} \rbrack}{\delta ( {i - k} )}} + {4{e_{ic}(t)}{e_{jc}( {t - \tau_{1}} )}^{ɛ}{e_{kc}( {t - \tau_{2}} )}}}}} & (36)\end{matrix}$

[0110] For cyclostationary sources, the matrices of third-order moments,T_(mcε)(t, τ₁, τ₂) and T_(xε)(t, τ₁, τ₂), are (quasi-) or poly-periodicfunctions of the time t accepting a Fourier series decomposition and weget: $\begin{matrix}{{T_{xɛ}( {t,\tau_{1},\tau_{2}} )} = {\sum\limits_{v_{xɛ} \in \Gamma_{x}^{\lbrack{1,{ɛ1}}\rbrack}}\quad {{T_{xɛ}^{v_{xɛ}}( {\tau_{1},\tau_{2}} )}^{{j2\pi}\quad v_{xɛ}t}}}} & (37)\end{matrix}$

[0111] where Γ_(x)^([1, ɛ1]) = {v_(xɛ)}

[0112] is the set of the cyclic frequencies v_(xε)of T_(xε)(t, τ₁, τ₂)and T_(xɛ)^(v_(xɛ))(τ₁, τ₂)

[0113] a matrix of cyclic third-order moments of x(t), definedrespectively by $\begin{matrix}{{T_{xɛ}^{v_{xɛ}}( {\tau_{1},\tau_{2}} )} = {{< {{T_{xɛ}( {t,\tau_{1},\tau_{2}} )}^{{- {j2\pi}}\quad v_{xɛ}t}} >_{c}} = {A\quad {T_{mcɛ}^{v_{xɛ}}( {\tau_{1},\tau_{2}} )}( {A \otimes A^{ɛ}} )^{ɛ\quad T}}}} & (38)\end{matrix}$

[0114] where {circle over (x)} corresponds to the Kronecker product.

[0115] Fourth-Order Statistics

[0116] Based on the above assumptions (relating to non-centered andcyclostationary signals) the fourth-order statistics of the observationsare the fourth-order cumulants defined by: $\begin{matrix}{{{Q\quad x\quad {{\zeta ( {t,\tau_{1},\tau_{2},\tau_{3}} )}\lbrack {i,j,k,l} \rbrack}}\overset{\Delta}{=}{{{Cum}( {{x_{i}(t)},{x_{j}( {t - \tau_{1}} )}^{\zeta 1},{x_{k}( {t - \tau_{2}} )}^{\zeta 2},{x_{l}( {t - \tau_{3}} )}} )} = {{{Cum}( {{{\Delta x}_{i}(t)},{\Delta \quad {x_{j}( {t - \tau_{1}} )}^{\zeta 1}},{\Delta \quad {x_{k}( {t - \tau_{2}} )}^{\zeta 2}},{\Delta \quad {x_{l}( {t - \tau_{3}} )}}} )} = {{{E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{j}( {t - \tau_{1}} )}^{\zeta 1}\Delta \quad {x_{k}( {t - \tau_{2}} )}^{\zeta 2}\Delta \quad {x_{l}( {t - \tau_{3}} )}} \rbrack} - {{E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{j}( {t - \tau_{1}} )}^{\zeta 1}} \rbrack}{E\lbrack {\Delta \quad {x_{k}( {t - \tau_{2}} )}^{\zeta 2}\Delta \quad {x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{k}( {t - \tau_{2}} )}^{\zeta 2}} \rbrack}{E\lbrack {\Delta \quad {x_{j}( {t - \tau_{1}} )}^{\zeta 1}\Delta \quad {x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{E\lbrack {\Delta \quad {x_{i}(t)}\Delta \quad {x_{l}( {t - \tau_{3}} )}} \rbrack}{E\lbrack {\Delta \quad {x_{j}( {t - \tau_{1}} )}^{\zeta 1}\Delta \quad {x_{k}( {t - \tau_{2}} )}^{\zeta 2}} \rbrack}}} = {{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}{x_{l}( {t - \tau_{3}} )}} \rbrack} - {{e_{xi}(t)}{E\lbrack {{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}{x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{e_{xj}( {t - \tau_{1}} )}^{\zeta 1}{E\lbrack {{x_{i}(t)}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}{x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{e_{xk}( {t - \tau_{2}} )}^{\zeta 2}{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{e_{xl}( {t - \tau_{3}} )}{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}} \rbrack}} - {{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{\zeta 1}} \rbrack}{E\lbrack {{x_{k}( {t - \tau_{2}} )}^{\zeta 2}{x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{E\lbrack {{x_{i}(t)}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}} \rbrack}{E\lbrack {{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{l}( {t - \tau_{3}} )}} \rbrack}} - {{E\lbrack {{x_{i}(t)}{x_{l}( {t - \tau_{3}} )}} \rbrack}{E\lbrack {{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}} \rbrack}} + {2{e_{xi}(t)}}}}}}}{{{e_{xj}( {t - \tau_{1}} )}^{\zeta 1}{E\lbrack {{x_{k}( {t - \tau_{2}} )}^{\zeta 2}{x_{l}( {t - \tau_{3}} )}} \rbrack}} + {2{e_{xi}(t)}{e_{xk}( {t - \tau_{2}} )}^{\zeta 2}{E\lbrack {{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{l}( {t - \tau_{3}} )}} \rbrack}} + \quad {2{e_{xi}(t)}{e_{xl}( {t - {\tau_{3}{E\lbrack {{{x_{j}( {t - {\tau 1}} )}^{\zeta 1}x_{k}} + ( {t - \tau_{2}} )^{\zeta 2}} \rbrack}} + {2{e_{xk}( {t - \tau_{2}} )}^{\zeta 2}{e_{xj}( {t - \tau_{1}} )}^{\zeta 1}{E\lbrack {{x_{i}(t)}{x_{l}( {t - \tau_{3}} )}} \rbrack}} + {2{e_{xl}( {t - \tau_{3}} )}{e_{xj}( {t - \tau_{1}} )}^{\zeta 1}{E\lbrack {{x_{k}( {t - \tau_{2}} )}^{\zeta 2}{x_{i}(t)}} \rbrack}} + {2{e_{xk}( {t - \tau_{2}} )}^{\zeta 2}{e_{xl}( {t - \tau_{3}} )}{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{\zeta 1}} \rbrack}} - {6\quad {e_{xi}(t)}{e_{xj}( {t - \tau_{1}} )}^{\zeta 1}{e_{xk}( {t - \tau_{2}} )}^{\zeta 2}{e_{xl}( {t - \tau_{3}} )}}} }}}} & (39)\end{matrix}$

[0117] where ζ ^(Δ) (ζ₁, ζ₂) and (ζ₁, ζ₂)=(1, 1), (−1, 1) or (−1, −1).Assuming that the term Q_(xζ)(t, τ₁, τ₂, τ₃)[i, j, k, l] is the element[N(i−1)+j, N(k−1)+l] of the matrix Q_(xζ)(t, τ₁, τ₂, τ₃), known as thequadricovariance matrix, with a dimension (N²×N²), we obtain anexpression of this dimension given by: $\begin{matrix}{{{Q\quad}_{x\quad \zeta}( {t,\tau_{1},\tau_{2},\tau_{3}} )} = {{( {A \otimes A^{\zeta 1}} ){Q_{m\quad c\quad \zeta}( {t,\tau_{1},\tau_{2},\tau_{3}} )}( {A^{\zeta 2} \otimes A} )^{T}} = {\sum\limits_{i,j,{k = 1}}\quad {{Q_{mc\zeta}\lbrack {i,j,k,l} \rbrack}{{( {t,\tau_{1},\tau_{2},\tau_{3}} )\lbrack {a_{i.} \otimes a_{j}^{\zeta 1}} \rbrack}\lbrack {a_{k}^{\zeta 2} \otimes a_{l}} \rbrack}^{T}}}}} & (40)\end{matrix}$

[0118] where Q_(mcζ)(t, τ₁, τ₂, τ₃) is the quadricovariance of thevector m_(c)(t) whose elements are Q_(mcζ)[i, j, k, l](t, τ₁, τ₂, τ₃)^(Δ) Cum(m_(ic)(t), m_(jc)(t−τ₁)^(ζ1), m_(kc)(t−τ₂)^(ζ2), m_(lc)(t−τ₃)).

[0119] For cyclostationary sources, the matrices of quadricovariance,Q_(mcζ)(t, τ₁, τ₂, τ₃) and Q_(xζ)(t, τ₁, τ₂, τ₃), are (quasi-) orpoly-periodic functions of the time t accepting a Fourier seriesdecomposition, and we obtain $\begin{matrix}{{{Q\quad}_{x\quad \zeta}( {t,\tau_{1},\tau_{2},\tau_{3}} )} = {\sum\limits_{\beta_{x\zeta} \in \Gamma_{x}^{\lbrack{1,{\zeta 1},{\zeta 2},1}\rbrack}}{{Q\quad}_{x\quad \zeta}^{\beta_{x\zeta}}( {\tau_{1},\tau_{2},\tau_{3}} )e^{{j2\pi}\quad \beta_{x\zeta}t}}}} & (41)\end{matrix}$

[0120] where Γ_(x)^([1, ζ1, ζ2, 1]) = {β_(xζ)}

[0121] is the set of cyclic frequencies β_(ζ)of Q_(xζ)(t, τ₁, τ₂, τ₃)and Q_(xζ)^(β_(xζ))(τ  ₁, τ  ₂, τ  ₃)

[0122] is a matrix of cyclic quadricovariance of x(t), definedrespectively by $\begin{matrix}{{Q_{x\quad \zeta}^{\beta_{x\quad \zeta}}( {{\tau \quad}_{1},{\tau \quad}_{2},{\tau \quad}_{3}} )} = {{< {{Q_{x\quad \zeta}( {{\tau \quad}_{1},{\tau \quad}_{2},{\tau \quad}_{3}} )}^{{- {j2\pi}}\quad \beta_{x\quad \zeta}t}} >_{c}} = {( {A \otimes A^{\zeta 1}} ){Q_{m\quad c}^{\beta_{x}}( {{\tau \quad}_{1},{\tau \quad}_{2},{\tau \quad}_{3}} )}( {A^{\zeta 2} \otimes A} )^{T}}}} & (42)\end{matrix}$

[0123] In particular, the cyclic quadricovariance for the zero cyclicfrequency corresponds to the temporal mean in t, Q_(xζ)(τ₁, τ₂, τ₃), deQ_(xζ)(t, τ₁, τ₂, τ₃), which is written as follows:

Q _(xζ)(τ₁, τ₂, τ₃) ^(Δ) <Q _(xζ)(t, τ₁, τ₂, τ₃)>_(c)=(A{circle over(x)}A ^(ζ1))Q _(mcζ)(τ₁, τ₂, τ₃)(A ^(ζ2) {circle over (x)}A)^(T)  (42b)

[0124] where Q_(mcζ)(τ₁, τ₂, τ₃) is the temporal mean in t of Q_(mcζ)(t,τ₁, τ₂, τ₃).

[0125] Novel Estimator of the Fourth-Order Cumulants of the Observations

[0126] From (39) and from the Fourier series decomposition of thestatistics appearing in this expression, we deduce the expression of theelement [i, j, k, l ] of the matrix of cyclic quadricovarianceQ_(xζ)^(β)(τ  ₁, τ  ₂, τ  ₃)

[0127] in the general case of non-centered observations, given by:$\begin{matrix}{{{{{M_{x\zeta}^{\beta}( {{\tau \quad}_{1},{\tau \quad}_{2},{\tau \quad}_{3}} )}\lbrack {i,j,k,l} \rbrack} - {\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}\quad \{ {{{e_{x}^{\gamma}\lbrack i\rbrack}{{T_{x\zeta}^{\beta - \gamma}( {{{\tau \quad}_{1} - {\tau \quad}_{3}},{\tau_{2} - {\tau \quad}_{3}}} )}\lbrack {l,j,k} \rbrack}^{{- {j2\pi}}{({\beta - \gamma})}\tau_{3}}} + {{e_{x}^{\gamma}\lbrack j\rbrack}^{\zeta 1}{{T_{x\zeta 2}^{\beta - {\gamma\zeta 1}}( {\tau_{2},\tau_{3}} )}\lbrack {i,k,l} \rbrack}^{- {j2\pi\gamma\zeta 1\tau}_{1}}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{\zeta 2}{{T_{x\zeta 1}^{\beta - {\gamma\zeta 2}}( {\tau_{1},\tau_{3}} )}\lbrack {i,j,l} \rbrack}^{- {j2\pi\gamma\zeta 2\tau}_{2}}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{{T_{x\zeta}^{\beta - \gamma}( {\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}^{- {j2\pi\gamma\tau}_{3}}}} \}} - {\sum\limits_{\alpha \in \Gamma_{x}^{\lbrack{1,{ɛ1}}\rbrack}}\quad {{{R_{x\zeta 1}^{\alpha}( \tau_{1} )}\lbrack {i,j} \rbrack}{{R_{x\zeta 2}^{{\zeta 2}{({\beta - \alpha})}}( {\tau_{3} - \tau_{2}} )}\lbrack {k,l} \rbrack}^{\zeta 2}^{{{j2\pi}{({\alpha - \beta})}}\tau_{2}}}} - {\sum\limits_{\gamma \in \Gamma_{x}^{\lbrack{1,{ɛ2}}\rbrack}}\quad {{{R_{x\zeta 2}^{\gamma}( \tau_{2} )}\lbrack {i,k} \rbrack}{{R_{x\zeta 1}^{{\zeta 1}{({\beta - \gamma})}}( {\tau_{3} - \tau_{1}} )}\lbrack {j,l} \rbrack}^{\zeta 1}^{{{j2\pi}{({\gamma - \beta})}}\tau_{1}}}} - {\sum\limits_{\omega \in \Gamma_{x}^{\lbrack{1,1}\rbrack}}\quad {{{R_{x\zeta 1}^{\omega}( \tau_{3} )}\lbrack {i,l} \rbrack}{{R_{x\zeta 12}^{{\zeta 1}{({\beta - \omega})}}( {\tau_{2} - \tau_{1}} )}\lbrack {j,k} \rbrack}^{\zeta 1}^{{{j2\pi}{({\omega - \beta})}}\tau_{1}}}} + {2{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\quad}\{ {{{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{{R_{x\zeta 2}^{\beta - \gamma - {\omega\zeta 1}}( {\tau_{2} - \tau_{3}} )}\lbrack {l,k} \rbrack}^{{j2\pi\omega\zeta 1}{({\tau_{3} - \tau_{1}})}}^{{{j2\pi}{({\gamma - \beta})}}\tau_{3}}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack k\rbrack}^{\zeta 2}{{R_{x\zeta 1}^{\beta - \gamma - {\omega\zeta 2}}( {\tau_{1} - \tau_{3}} )}\lbrack {l,j} \rbrack}^{{j2\pi\omega\zeta 2}{({\tau_{3} - \tau_{2}})}}^{{{j2\pi}{({\gamma - \beta})}}\tau_{3}}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack l\rbrack}{{R_{x\zeta 12}^{{\zeta 1}{({\beta - \gamma - \omega})}}( {\tau_{2} - \tau_{1}} )}\lbrack {j,k} \rbrack}^{\zeta 1}^{{j2\pi\omega}{({\tau_{1} - \tau_{3}})}}^{- {j2\pi\beta\tau}_{1}}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{\zeta 2}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{{R_{x1}^{\beta - {\gamma\zeta 2} - {\omega\zeta 1}}( \tau_{3} )}\lbrack {i,l} \rbrack}^{{- {j2\pi\gamma\tau}_{2}}{\zeta 2}}^{{- {j2\pi\omega\tau}_{1}}{\zeta 1}}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{{R_{x\quad {\zeta 2}}^{\beta - \gamma - {\omega\zeta 1}}( \tau_{2} )}\lbrack {i,k} \rbrack}^{{- {j2\pi\omega\tau}_{1}}{\zeta 1}}^{- {j2\pi\gamma\tau}_{3}}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{e_{x}^{\omega}\lbrack k\rbrack}^{\zeta 2}{{R_{x\quad {\zeta 1}}^{\beta - \gamma - {\omega\zeta 2}}( \tau_{1} )}\lbrack {i,j} \rbrack}^{{- {j2\pi\omega\tau}_{2}}{\zeta 2}}^{- {j2\pi\gamma\tau}_{3}}}} \}}}} - {6{\sum\limits_{\underset{^{{j2\pi\tau}_{3}{({\gamma - \beta})}}}{\gamma \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\delta \in \Gamma_{x}^{1}}\quad}{{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{\zeta 1}{e_{x}^{\delta}\lbrack k\rbrack}^{\zeta 2}{e_{x}^{\beta - {\omega\zeta 1} - {\delta\zeta 2}}\lbrack l\rbrack}^{{j2\pi\omega\zeta 1}{({\tau_{3} - \tau_{1}})}}^{{j2\pi\delta\zeta 2}{({\tau_{3} - \tau_{2}})}}}}}}}}\quad \quad {{{{where}\quad {{T_{x\zeta}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}}\overset{\Delta}{=}{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{\zeta 1}{x_{k}( {t - \tau_{2}} )}^{\zeta 2}} \rbrack}},{{{T_{x\zeta}^{\beta}( {\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}\overset{\Delta}{=}{< {{{T_{x\zeta}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}^{{- {j2\pi\beta}}\quad t}} >_{c}}},\quad {{{T_{xɛ}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}\overset{\Delta}{=}{E\lbrack {{x_{i}(t)}{x_{j}( {t - \tau_{1}} )}^{ɛ}{x_{k}( {t - \tau_{2}} )}} \rbrack}},\quad {{{T_{xɛ}^{\beta}( {\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack} = {< {{{T_{xɛ}( {t,\tau_{1},\tau_{2}} )}\lbrack {i,j,k} \rbrack}\quad ^{{- {j2\pi\beta}}\quad t}} >_{c}}}}}\quad} & (43)\end{matrix}$

[0128] In particular, the matrix of cyclic quadricovariance exploited bythe separators of the family F3 corresponds to the matrix Q_(xζ)^(β)(τ₁, τ₂, τ₃) for β=0ζ=(−1, −1) and (τ₁, τ₂, τ₃)=(0, 0, 0) and itselement Q_(xζ) ^(β)(τ₁, τ₂, τ₃)[i, j, k, l], denoted as Q_(x)[i, j, k,l], is written: $\begin{matrix}{{{Q_{x}\lbrack {i,j,k,l} \rbrack}\overset{\Delta}{=}{{< {{Q_{x}( {t,0,0,0} )}\lbrack {i,j,k,l} \rbrack} >_{c}} = {{M_{x}^{0}\lbrack {i,j,k,l} \rbrack} - {\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}\quad \{ {{{e_{x}^{\gamma}\lbrack i\rbrack}{T_{x}^{\gamma}\lbrack {j,l,k} \rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{T_{x}^{\gamma}\lbrack {j,i,k} \rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack j\rbrack}^{*}{T_{x}^{\gamma}\lbrack {i,k,l} \rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{*}{T_{x}^{\gamma}\lbrack {i,j,l} \rbrack}}} \}} - {\sum\limits_{\alpha \in \Gamma_{x}^{\lbrack{1,{- 1}}\rbrack}}\{ {{{R_{x}^{\alpha}\lbrack {i,j} \rbrack}{R_{x}^{- \alpha}\lbrack {l,k} \rbrack}} + {{R_{x}^{\alpha}\lbrack {i,k} \rbrack}{R_{x}^{- \alpha}\lbrack {l,j} \rbrack}}} \}} - {\sum\limits_{\omega \in \Gamma_{x}^{\lbrack{1,1}\rbrack}}\quad {{C_{x}^{\omega}\lbrack {i,l} \rbrack}{C_{x}^{\omega}\lbrack {j,k} \rbrack}^{*}}} + {2{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}{\underset{{\omega \in \Gamma_{x}^{1}}\quad}{\sum\{}{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{R_{x}^{\omega - \gamma}\lbrack {l,k} \rbrack}}}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack k\rbrack}^{*}{R_{x}^{\omega - \gamma}\lbrack {l,j} \rbrack}} + {{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack l\rbrack}{C_{x}^{\omega + \gamma}\lbrack {k,j} \rbrack}^{*}} + {{e_{x}^{\gamma}\lbrack k\rbrack}^{*}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{C_{x}^{\omega + \gamma}\lbrack {i,l} \rbrack}} + {{e_{x}^{\gamma}\lbrack l\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{R_{x}^{\omega - \gamma}\lbrack {i,k} \rbrack}} + {e_{x}^{\gamma}\lbrack {l\quad 1{e_{x}^{\omega}\lbrack k\rbrack}^{*}{R_{x}^{\omega - \gamma}\lbrack {i,j} \rbrack}} \}} - {6{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\delta \in \Gamma_{x}^{1}}\quad}{{e_{x}^{\gamma}\lbrack i\rbrack}{e_{x}^{\omega}\lbrack j\rbrack}^{*}{e_{x}^{\delta}\lbrack k\rbrack}^{*}{e_{x}^{\delta + \omega - \gamma}\lbrack l\rbrack}}}}}}}}}{{{{where}\quad {M_{x}^{0}\lbrack {i,j,k,l} \rbrack}}\overset{\Delta}{=}{< {E\lbrack {{x_{i}(t)}{x_{j}(t)}^{*}{x_{k}(t)}^{*}{x_{l}(t)}} \rbrack} >_{c}}},{{T_{x}^{\beta}\lbrack {i,j,k} \rbrack}\overset{\Delta}{=}{< {{E\lbrack {{x_{i}(t)}{x_{j}(t)}^{*}{x_{k}(t)}} \rbrack}^{{- {j2\pi\beta}}\quad t}} >_{c}}},{{R_{x}^{\alpha}\lbrack {i,j} \rbrack}\overset{\Delta}{=}{< {{E\lbrack {{x_{i}(t)}{x_{j}(t)}^{*}} \rbrack}^{{- {j2\pi\alpha}}\quad t}} >_{c}}},{{C_{x}^{\alpha}\lbrack {i,j} \rbrack}\overset{\Delta}{=}{< {{E\lbrack {{x_{i}(t)}{x_{j}(t)}} \rbrack}^{{- {j2\pi\alpha}}\quad t}} >_{c}.}}}} & (44)\end{matrix}$

[0129] Thus, the estimation of the matrix of cyclic quadricovarianceQ_(x  ζ)^(β)(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l]

[0130] l₂T_(e), l₃T_(e))[i, j, k, l] is made on the basis of theestimator Q̂_(x)^(β)ζ(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l](K)

[0131] l](K) defined by: $\begin{matrix}{{{{{\hat{Q}}_{x_{\zeta}}^{\beta}( {{l_{1}T_{e}},{l_{2}T_{e}},{l_{3}T_{e}}} )}\lbrack {i,j,k,l} \rbrack}(K)} = {{{{{\hat{M}}_{x_{\zeta}}^{\beta}( {{l_{1}T_{e}},{l_{2}T_{e}},{l_{3}T_{e}}} )}\lbrack {i,j,k,l} \rbrack}(K)} - {\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{{\hat{T}}_{x_{\zeta}}^{\beta - \gamma}( {{( {l_{1} - l_{3}} )T_{e}},{( {l_{2} - l_{3}} )T_{e}}} )}\lbrack {l,j,k} \rbrack}(K)^{{- {{j2\pi}{({\beta \quad - \gamma})}}}l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{T}}_{x_{\zeta 2}}^{\beta - {\gamma\zeta 1}}( {{l_{2}T_{e}},{l_{3}T_{e}}} )}\lbrack {i,k,l} \rbrack}(K)^{{- {j2\pi\gamma\zeta 2l}_{1}}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{\zeta 2}{{{\hat{T}}_{x_{\zeta 2}}^{\beta - {\gamma\zeta 2}}( {{l_{1}T_{e}},{l_{3}T_{e}}} )}\lbrack {i,j,l} \rbrack}(K)^{{- {j2\pi\gamma\zeta}_{2}}l_{2}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{{\hat{T}}_{x_{\zeta}}^{\beta - \gamma}( {{l_{1}T_{e}},{l_{2}T_{e}}} )}\lbrack {i,j,k} \rbrack}(K)^{{- {j2\pi\gamma}}\quad l_{3}T_{e}}}} \}} - {\sum\limits_{\alpha \in \Gamma_{x}^{\lbrack{1,{ɛ1}}\rbrack}}{{{{\hat{R}}_{x_{\zeta 1}}^{\alpha}( {l_{1}T_{e}} )}\lbrack {i,j} \rbrack}(K){{{\hat{R}}_{x_{\zeta 2}}^{{\zeta 2}{({\beta - \alpha})}}( {( {l_{3} - l_{2}} )T_{e}} )}\lbrack {k,l} \rbrack}(K)^{\zeta 2}^{{{j2\pi}{({\alpha - \beta})}}l_{2}T_{e}}}} - {\sum\limits_{\gamma \in \Gamma_{x}^{\lbrack{1,{ɛ2}}\rbrack}}{{{{\hat{R}}_{x_{\zeta 2}}^{\gamma}( {l_{2}T_{e}} )}\lbrack {i,k} \rbrack}(K){{{\hat{R}}_{x_{\zeta 1}}^{{\zeta 1}{({\beta - \gamma})}}( {( {l_{3} - l_{1}} )T_{e}} )}\lbrack {j,l} \rbrack}(K)^{\zeta 1}^{{{j2\pi}{({\gamma - \beta})}}l_{1}T_{e}}}} - {\sum\limits_{\omega \in \Gamma_{x}^{\lbrack{1,1}\rbrack}}{{{{\hat{R}}_{x_{\zeta 1}}^{\omega}( {l_{3}T_{e}} )}\lbrack {i,l} \rbrack}(K){{{\hat{R}}_{x_{\zeta 2}}^{{\zeta 1}{({\beta - \omega})}}( {( {l_{2} - l_{1}} )T_{e}} )}\lbrack {j,k} \rbrack}(K)^{\zeta 1}^{{{j2\pi}{({\omega - \beta})}}l_{1}T_{e}}}} + {2{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\quad}\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{R}}_{x_{\zeta 2}}^{\beta - \gamma - \omega^{\zeta 1}}( {( {l_{2} - l_{3}} )T_{e}} )}\lbrack {l,k} \rbrack}(K)^{{{j2\pi\omega\zeta 1}{({l_{3} - l_{1}})}}T_{e}}^{{{j2\pi}{({\gamma - \beta})}}l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{\zeta 2}{{{\hat{R}}_{x_{\zeta 1}}^{\beta - \gamma - \omega^{\zeta 2}}( {( {l_{1} - l_{3}} )T_{e}} )}\lbrack {l,j} \rbrack}(K)^{{{j2\pi\omega\zeta 2}{({l_{3} - l_{2}})}}T_{e}}^{{{j2\pi}{({\gamma - \beta})}}l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack l\rbrack}(K)^{\zeta 2}{{{\hat{R}}_{x_{\zeta 2}}^{{\zeta 1}{({\beta - \omega - \gamma})}}( {( {l_{2} - l_{1}} )T_{e}} )}\lbrack {j,k} \rbrack}(K)^{\zeta 1}^{{{j2\pi\omega}{({l_{1} - l_{3}})}}T_{e}}^{{- {j2\pi\beta}}\quad l_{1}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{\zeta 2}{{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{R}}_{x_{\zeta 1}}^{\beta - {\gamma\zeta 2} - {\omega\zeta 1}}( {l_{3}T_{e}} )}\lbrack {i,l} \rbrack}(K)^{\quad^{{- {j2\pi\gamma}}\quad l_{2}T_{e}{\zeta 2}}}^{{- {j2\pi\omega}}\quad l_{1}T_{e}{\zeta 1}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{{\hat{R}}_{x_{\zeta 2}}^{\beta - \gamma - {\omega\zeta 1}}( {l_{2}T_{e}} )}\lbrack {i,k} \rbrack}(K)^{\quad^{{- {j2\pi\gamma}}\quad l_{1}T_{e}{\zeta 1}}}^{{- {j2\pi\gamma}}\quad l_{3}T_{e}}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{\zeta 2}{{{\hat{R}}_{x_{\zeta 1}}^{\beta - \gamma - {\omega\zeta 2}}( {l_{1}T_{e}} )}\lbrack {i,j} \rbrack}(K)^{\quad^{{- {j2\pi\omega}}\quad l_{2}T_{e}{\zeta 2}}}^{{- {j2\pi\gamma}}\quad l_{3}T_{e}}}} \}}}} - {6{\sum\limits_{{\gamma \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\omega \in \Gamma_{x}^{1}}\quad}{\sum\limits_{{\delta \in \Gamma_{x}^{1}}\quad}{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{\zeta 1}{{\hat{e}}_{x}^{\delta}\lbrack k\rbrack}(K)^{\zeta 2}{{\hat{e}}_{x}^{\beta - \gamma - {\omega\zeta 1} - {\delta\zeta 2}}\lbrack l\rbrack}(K)^{{{j2\pi\omega\zeta 1}{({l_{3} - l_{1}})}}T_{e}}^{{{j2\pi\delta\zeta 2}{({l_{3} - l_{2}})}}T_{e}}^{{j2\pi}\quad l_{3}{T_{e}{({\gamma - \beta})}}}}}}}}}} & (45)\end{matrix}$

[0132] where M̂_(x)^(β)ζ(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l](K)

[0133] is defined by the expression: $\begin{matrix}{{{{{\hat{M}}_{x\quad \zeta}^{\beta}( {{l_{1}T_{e}},{l_{2}T_{e}},{l_{3}T_{e}}} )}\lbrack {i,j,k,l} \rbrack}(K)} = {\frac{1}{K}{\sum\limits_{m = 1}^{K}{{x_{i}(m)}{x_{j}( {m - l_{1}} )}{{{}_{}^{\zeta 1}{}_{}^{}}( {m - l_{2}} )}{{{}_{}^{\zeta 2}{}_{}^{}}( {m - l_{3}} )}^{{- j}\quad 2\pi \quad \beta \quad m\quad T_{e}}}}}} & (46)\end{matrix}$

[0134] given in the reference [5] and whereê_(x)^(γ)[i](K), R̂_(x  ɛ)^(α)(lT_(e))[i, j](K), T̂_(x  ɛ)^(α)(l₁T_(e), l₂T_(e))[i, j, l](K)  and  T̂_(x  ζ)^(α)(l₁T_(e), l₂T_(e))[i, j, l](K)  

[0135] are defined respectively by (47), (48), (49) and (50).$\begin{matrix}{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\quad \frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}^{{- {j2\pi}}\quad \gamma \quad k\quad T_{e}}}}} & (47) \\{{{{\hat{R}}_{x\quad ɛ}^{\alpha}( {lT}_{e} )}\lbrack {i,j} \rbrack}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\quad \frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{x_{j}( {k - l} )}{{}_{}^{}{}_{}^{{- j}\quad 2\pi \quad \alpha \quad k\quad T_{e}}}}}} & (48) \\{{{{\hat{T}}_{x\quad ɛ}^{\alpha}( {{l_{1}T_{e}},{l_{2}T_{e}}} )}\lbrack {i,j,l} \rbrack}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\quad \frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{x_{j}( {k - l_{1}} )}{{\,^{ɛ}\quad x_{l}}( {k - l_{2}} )}^{{- j}\quad 2\pi \quad \alpha \quad {kT}_{e}}}}} & (49) \\{{{{\hat{T}}_{x\quad \zeta}^{\alpha}( {{l_{1}T_{e}},{l_{2}T_{e}}} )}\lbrack {i,j,l} \rbrack}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\quad \frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{i}(k)}{x_{j}( {k - l_{1}} )}{{{}_{}^{\zeta 1}{}_{}^{}}( {k - l_{2}} )}{{}_{}^{\zeta 2}{}_{}^{{- j}\quad 2\pi \quad \alpha \quad k\quad T_{e}}}}}} & (50)\end{matrix}$

[0136] Thus, assuming cyclostationary and cycloergodic observations,centered or non-centered, the estimator (45) is an asymptoticallyunbiased and consistent estimator of the elementQ_(x  ζ)^(β)(l₁T_(e), l₂T_(e), l₃T_(e))[i, j, k, l].

[0137] In particular, the separators F3 must exploit the estimator (45)for β=0, ζ=(−1, −1) and l₁=l₂=l₃=0, denoted {circumflex over (Q)}_(x)[i,j, k, l](K) and defined by: $\begin{matrix}{{{{\hat{Q}}_{x}\lbrack {i,j,k,l} \rbrack}(K)\quad \underset{\underset{\_}{\_}}{\Delta}\quad {{\hat{M}}_{x}^{0}\lbrack {i,j,k,l} \rbrack}(K)} - {\sum\limits_{\gamma \quad \in \quad \Gamma_{x}^{1}}\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{T}}_{x}^{\gamma}\lbrack {j,l,k} \rbrack}(K)^{*}} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{T}}_{x}^{\gamma}\lbrack {j,i,k} \rbrack}(K)^{*}} + {{{\hat{e}}_{x}^{\gamma}\lbrack j\rbrack}(K)^{*}{{\hat{T}}_{x}^{\gamma}\lbrack {i,k,l} \rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{*}{{\hat{T}}_{x}^{\gamma}\lbrack {i,j,l} \rbrack}(K)}} \}} - {\sum\limits_{\omega \quad \in \quad \Gamma_{x}^{\lbrack{1,1}\rbrack}}{{{\hat{C}}_{x}^{\omega}\lbrack {i,l} \rbrack}(K){{\hat{C}}_{x}^{\omega}\lbrack {j,k} \rbrack}(K)^{*}}} - {\sum\limits_{\alpha \quad \in \quad \Gamma_{x}^{\lbrack{1,{- 1}}\rbrack}}\{ {{{{\hat{R}}_{x}^{\alpha}\lbrack {i,j} \rbrack}(K){{\hat{R}}_{x}^{- \alpha}\lbrack {l,k} \rbrack}(K)} + {{{\hat{R}}_{x}^{\alpha}\lbrack {i,k} \rbrack}(K){{\hat{R}}_{x}^{- \alpha}\lbrack {l,j} \rbrack}(K)}} \}} + {2{\sum\limits_{\gamma \quad \in \quad \Gamma_{x}^{1}}{\sum\limits_{\omega \quad \in \quad \Gamma_{x}^{1}}\{ {{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\lbrack {l,k} \rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\lbrack {l,j} \rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack l\rbrack}(K){{\hat{C}}_{x}^{\omega + \gamma}\lbrack {k,j} \rbrack}(K)^{*}} + {{{\hat{e}}_{x}^{\gamma}\lbrack k\rbrack}(K)^{*}{{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{C}}_{x}^{\omega + \gamma}\lbrack {i,l} \rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\lbrack {i,k} \rbrack}(K)} + {{{\hat{e}}_{x}^{\gamma}\lbrack l\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack k\rbrack}(K)^{*}{{\hat{R}}_{x}^{\omega - \gamma}\lbrack {i,j} \rbrack}(K)}} \}}}} - {6{\sum\limits_{\gamma \quad \in \quad \Gamma_{x}^{1}}{\sum\limits_{\omega \quad \in \quad \Gamma_{x}^{1}}{\sum\limits_{\delta \quad \in \quad \Gamma_{x}^{1}}{{{\hat{e}}_{x}^{\gamma}\lbrack i\rbrack}(K){{\hat{e}}_{x}^{\omega}\lbrack j\rbrack}(K)^{*}{{\hat{e}}_{x}^{\delta}\lbrack k\rbrack}(K)^{*}{{\hat{e}}_{x}^{\delta + \omega - \gamma}\lbrack l\rbrack}(K)}}}}}} & (51)\end{matrix}$

[0138] where {circumflex over (M)}_(x) ⁰[i, j, k, l](K) is defined by(46) (defined here above) with β=0, l₁=l₂=l₃=0, ζ=(−1, −1), {circumflexover (T)}_(x) ^(γ)[i, j, l](K) is defined by (49) with α=γ, l₁=l₂=0,ε=−1, {circumflex over (R)}_(x) ^(α)[i, j] and Ĉ_(x) ^(α)[i, j] aredefined by (29) with l=0 and respectively ε=−1 and ε=+1.

[0139] As indicated here above and as can be seen from the previousestimators, the method according the invention also comprises a step inwhich the second-order estimator is corrected by using cyclicfrequencies which have to be detected a priori. The following detailedexample is given in the case of a detection of the first-order cyclicfrequencies.

[0140] Detector of the First-Order Cyclic Frequencies

[0141] The detection of the first-order cyclic frequencies of theobservations γ by a detector of cyclic frequencies and the constitutionof an estimate, Γ̂_(x)¹,

[0142] of the set, Γ_(x)¹,

[0143] of the cyclic frequencies γ are done, for example, by computingthe following standardized criterion: $\begin{matrix}{{{V(\alpha)} = \frac{\frac{1}{N}{\sum\limits_{n = 1}^{N}{{{{\hat{e}}_{x}^{\alpha}\lbrack n\rbrack}\lbrack K\rbrack}}^{2}}}{{\hat{\gamma}}_{x}}}\quad {{With}\quad {\hat{\gamma}}_{x}} = {{\frac{1}{N}\frac{1}{K}{\sum\limits_{k = 1}^{K}{{{x_{n}(k)}}^{2}\quad {and}\quad {{\hat{e}}_{x}^{\gamma}\lbrack n\rbrack}(K)}}} = {\frac{1}{K}{\sum\limits_{k = 1}^{K}{{x_{n}(k)}\quad {\exp ( {{- j}\quad 2\quad \pi \quad \gamma \quad k\quad T_{e}} )}}}}}} & (52)\end{matrix}$

[0144] It may be recalled that x_(n)(t) is the signal received at then^(th) sensor. The estimator ê_(x) ^(γ)[n](K) may be computed in anoptimized way for K cyclic frequencies α_(k)=k/(K T_(e)) (0≦k≦K−1) by anFFT (a Fast Fourier Transformation) on the temporal samplesx_(n)((k+k₀)T_(e)) such that 0≦k≦K−1. The criterion V(α) is standardizedbetween 0 and 1 because {overscore (γ)}_(x) represents the temporal meanof the mean power of the signal on the set of sensors. It being knownthat, on the assumption that x_(n)(kT_(e)) is a Gaussian noise, thecriterion V(α) approximately follows a chi-2 relationship, the followingdetection test is deduced therefrom:

[0145] The frequency α is a cyclic frequency γ_(n) of E[x(t)]:V(α)>=μ(pfa)

[0146] The frequency α is not a cyclic frequency: V(α)<μ(pfa)

[0147] where μ(pfa) is a threshold as a function of the probability offalse alarm pfa

[0148] The rest of the description gives several alternative modes ofimplementation of the method for the separation of statisticallyindependent, stationary or cyclostationary sources. The associatedseparators are respectively called F′1, F′2, F′3 and F′4.

[0149] Proposed Second-Order Separators

[0150] Separators F1′

[0151] The separators of the family F1′ are self-learning second-orderseparators implementing the following operations:

[0152] Whitening Step

[0153] The detection of the first-order cyclic frequencies of theobservations y by any detector of cyclic frequencies and theconstitution of an estimate, Γ̂_(x)¹,

[0154] of the set, Γ_(x)¹,

[0155] of the cyclic frequencies γ.

[0156] The estimation of the matrix R_(Δx)(0) by {circumflex over(R)}_(Δx)(0)(K) defined by (33) and (32) for l=0 on the basis of a givennumber K of samples.

[0157] The detection of the number of sources P from the decompositionof {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All thenon-deterministic sources are detected).

[0158] The computation of the whitening matrix of the observations,{circumflex over (T)}, where {circumflex over (T)} ^(Δ) {circumflex over(Λ)}_(s) ^(−1/2) Û_(s) ^(†), with a dimension (P×N), where {circumflexover (Λ)}_(s) is the diagonal matrix (P×P) of the P greatest eigenvalues of {circumflex over (R)}_(Δx)(0)(K)−λmin I, λmin is the minimumeigen value of {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrixof associated eigen vectors. We write z(t) ^(Δ) {circumflex over (T)}x(t). (The directional vectors of the non-deterministic sources areorthonormalized).

[0159] Identification Step

[0160] Choice of Q values of non-zero delays, l_(q), (1≦q≦Q).

[0161] For each value, l_(q), of the delay, estimation of the matrix ofaveraged second-order cumulants of the observations, R_(Δx)(l_(q)T_(e)),by {circumflex over (R)}_(Δx)(l_(q)T_(e))(K) defined by (33), (29) and(32)

[0162] Computation of the matrices {circumflex over(R)}_(Δz)(l_(q)T_(e))(K) ^(Δ{circumflex over (T)}{circumflex over (R)})_(Δx)(l_(q)T_(e))(K){circumflex over (T)}^(†)and self-learnedidentification of the directional vectors of the whitened sources bymaximization, with respect to U ^(Δ) (u₁, u₂, . . . up), of thecriterion${{C1}(U)}\quad \underset{\underset{\_}{\_}}{\Delta}\quad {\sum\limits_{q = 1}^{Q}{\sum\limits_{l = 1}^{P}{{u_{l}^{\dagger}{R_{z}( \tau_{q} )}u_{l}}}^{2}}}$

[0163] given in the reference [7] where R_(z)(τ_(q)) is replaced by{circumflex over (R)}_(Δz)(l_(q)T_(e))(K). The solution matrix U isdenoted Â_(nd)′ and contains an estimate of the whitened directionalvectors of the non-deterministic sources.

[0164] Filter

[0165] The computation of an estimate of the matrix of the directionalvectors of the non-deterministic source Â_(nd)=Û_(s){circumflex over(Λ)}_(s) ^(1/2)Â_(nd)′

[0166] The extraction of the non-deterministic sources by any spatialfiltering of the observations constructed on the basis of Â_(nd).

[0167] Processing of the Deterministic Sources

[0168] The construction of the orthogonal projector on the spaceorthogonal to the columns of Â_(nd):Proj=I−Â_(nd)[Â_(nd†)Â_(nd)]⁻¹Â_(nd) ^(†)

[0169] The implementation of the SOBI algorithm [3] on the basis of theobservations v(t) ^(Δ) Proj x(t) to identify the directional vectors ofthe deterministic sources and extract them.

[0170] Separators F2′

[0171] The separators of the family F2′ are second-order self-learningseparators implementing the following operations:

[0172] Whitening Step

[0173] The detection of the first order cyclic frequencies of theobservations γ by any unspecified detector of cyclic frequencies and theconstitution of an estimate, Γ̂_(x)¹,

[0174] of the set, Γ_(x)¹,

[0175] of the cyclic frequencies γ.

[0176] The estimation of the matrix R_(Δx)(0) by {circumflex over(R)}_(Δx)(0)(K) defined by (33) and (32) for l=0 from a given number ofsamples K

[0177] The detection of the number of sources P from the decompositioninto eigen elements of {circumflex over (R)}_(Δx)(0)(K). (all thenon-deterministic sources are detected).

[0178] Identification Step

[0179] The computation of the whitening matrix of the observations,{circumflex over (T)}, where {circumflex over (T)} ^(Δ) {circumflex over(Λ)}_(s) ^(−1/2) Û_(s) ^(†), with a dimension (P×N), where {circumflexover (Λ)}_(s) is the diagonal matrix (P×P) of the P biggest eigen valuesof {circumflex over (R)}_(Δx)(0)(K)−λmin I, λmin is the minimal eigenvalue of {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of theassociated eigen vectors. We write z(t) ^(Δ) {circumflex over (T)} x(t).(The directional vectors of the non-deterministic sources areorthonormalized).

[0180] The detection of the second-order cyclic frequencies of theobservations [9], α_(ε), for ε=−1 and ε=+1 by any detector of cyclicfrequencies and constitution of the estimates,Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]),

[0181] the sets respectively, Γ_(x) ^([1, −1]) and Γ_(x) ^([1, 1]), thecyclic frequencies respectively of the first and second matrices ofcorrelation of the observations.

[0182] The choice of an arbitrary number of pairs (α_(m), ε_(m)) (1≦m≦M)such that, for each of these pairs, at least one source possesses thesecond-order cyclic frequency α_(m) for a matrix of correlation indexedby ε_(m).

[0183] For each pair (α_(m), ε_(m)), (1≦m≦M):

[0184] The choice of an arbitrary number Q_(m) of delays, l_(mq),(1≦q≦Q_(m))

[0185] For each value of the delay, l_(mq) (1≦q≦Q_(m)),

[0186] The estimation of the matrix of second-order cyclic cumulants ofthe whitened observations,R_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e)), by  R̂_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)

[0187] defined by (31), (29), (32) with the index z instead of x.

[0188] The detection of the number of non-deterministic sourcesP_((αm, εm)) having the second-order cyclic frequency α_(m), for thematrix of correlation indexed by ε_(m). This detection test may consistof a search for the maximum rank of the signal space of the matrices

[0189] {circumflex over (R)}_(Δzε), ^(αm) _(m)(l_(mq)T_(e))(K).Û_((αm, εm)) denotes the unit matrix (P×P_((α) _(m, εm))), obtained bySVD of the previous matrices, whose columns generate the space generatedby the whitened directional vectors associated with the sources havingthe second-order cyclic frequency, α_(m), for the matrix of correlationindexed by ε_(m).

[0190] Reduction of dimension: We write v(t) ^(Δ) Û_((αm, εm)) ^(†)z(t),with a dimension (P_((αm, εm))×1) and carry out a computation, for eachdelay l_(mq) (1≦q≦Q_(m)), of the matricesR̂_(Δ  v  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K) = Û_((α  m, ɛ  m))^(†)R̂_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)Û_((α  m, ɛ  m))^(*ɛ_(m))

[0191] The self-learned identification of the doubly whiteneddirectional vectors associated with the pair (α_(m), ε_(m)) bymaximization in relation to U ^(Δ) (u₁, u₂, . . . , up_((αm, εm))), ofthe criterion${{C1}(U)}\quad \underset{\underset{\_}{\_}}{\Delta}\quad {\sum\limits_{q = 1}^{Q}{\sum\limits_{l = 1}^{P}{{u_{l}^{\dagger}{R_{z}( \tau_{q} )}u_{l}}}^{2}}}$

[0192] given in the reference [7] where R_(z)(τ_(q)) is replaced byR̂_(Δ  v  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)R̂_(Δ  v  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)^(†).

[0193] The solution matrix U is written as Â_(nd) _((αm, εm)) ′ andcontains an estimate of the doubly whitened directional vectors of thenon-deterministic sources associated with the pair (α_(m), ε_(m)).

[0194] Filter

[0195] The computation of an estimate of the matrix of the directionalvectors of the non-deterministic sources associated with the pair(α_(m), ε_(m)): Â_(nd) _((αm, εm)) =Û_(s){circumflex over (Λ)}_(s)^(1/2)Û_((αm, εm))Â_(nd) _((αm, εm)) ′

[0196] The concatenation of the matrices Â_(nd) _((αm, εm)) for all thepairs (α_(m), ε_(m)), (1≦m≦M). We obtain the matrix (N×P) of Â_(nd) ofthe directional vectors of the non-deterministic sources.

[0197] The extraction of the non-deterministic sources by any spatialfiltering of the observations constructed from Â_(nd).

[0198] Processing of the Deterministic Sources

[0199] The construction of the orthogonal projector on the spaceorthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)^(†)Â_(nd)]⁻¹Â_(nd) ^(†)

[0200] The implementation of the SOBI algorithm [3] from theobservations w(t) ^(Δ) Proj x(t) to identify the directional vectors ofthe deterministic sources and extract them.

[0201] Proposed Fourth-Order Separators

[0202] Separators F3′

[0203] The separators of the family F3′ are fourth-order self-learningseparators implementing the following operations:

[0204] Whitening Step

[0205] The detection of the first-order cyclic frequencies of theobservations γ by any detector of cyclic frequencies and theconstitution of an estimate, Γ̂_(x)¹,

[0206] of the set, Γ_(x)¹,

[0207] of the cyclic frequencies γ.

[0208] The detection of the second-order cyclic frequencies of theobservations, α_(ε), for ε=−1 and ε=+1 by any detector of cyclicfrequencies and the constitution of the estimates,Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]),

[0209] of the sets respectively,

[0210] Γ_(x) ^([1, −1])and Γ̂_(x)^([1, 1]),

[0211] of the cyclic frequencies respectively of the first and secondmatrices of correlation of the observations.

[0212] The estimation of the matrix R_(Δx)(0) by {circumflex over(R)}_(Δx)(0)(K) defined by (53) and (52) for l=0 from a given number ofsamples K

[0213] The detection of the number of sources P from the decompositionof {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All thenon-deterministic sources are detected).

[0214] The computation of the whitening matrix of the observations,{circumflex over (T)}, where {circumflex over (T)} ^(Δ) {circumflex over(Λ)}_(s) ^(−1/2) Û_(s) ^(†), with a dimension (P×N), where {circumflexover (Λ)}_(s) is the diagonal matrix (P×P) of the P greatest eigenvalues of {circumflex over (R)}_(Δx)(0)(K)−λmin I, λmin is the minimumeigen value of {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrixof associated eigen vectors. We write z(t) ^(Δ) {circumflex over (T)}x(t). (The directional vectors of the non-deterministic sources areorthonormalized).

[0215] Identification Step

[0216] The estimation of the quadricovariance, Q_(z), of the vector z(t)by the expressions (51), (29), (49), (2) and (32) with the index zinstead of x.

[0217] The decomposition into eigen elements of {circumflex over(Q)}_(z) and the estimation of the P eigen matrices M_(zi) (1≦i≦P)associated with the P eigen values of higher-value moduli.

[0218] The joint diagonalization of the P eigen matrices M_(zi) weightedby the associated eigen values and the obtaining of the matrix of thewhitened directional vectors of the non-deterministic sources Â_(nd)′.

[0219] Filter

[0220] The computation of an estimate of the matrix of the directionalvectors of the non-deterministic sources Â_(nd)=Û_(s) Û_(s){circumflexover (Λ)}_(s) ^(1/2)Â_(nd)′

[0221] The extraction of the non-deterministic sources by any spatialfiltering of the observations constructed from Â_(nd).

[0222] Processing of the Deterministic Sources

[0223] The construction of the orthogonal projector on the spaceorthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)^(†)Â_(nd)]⁻¹Â_(nd) ^(†)

[0224] The implementation of the algorithm JADE [4] from theobservations v(t)ΔProj x(t) to identify the directional vectors of thedeterministic sources and extract them.

[0225] Separators F4′

[0226] The separators of the family F4′ are fourth-order self-learningseparators associated with the reference [8] implementing the followingoperations:

[0227] Whitening Step

[0228] The detection of the first-order cyclic frequencies of theobservations y by any detector of cyclic frequencies and theconstitution of an estimate Γ̂_(x)¹,

[0229] of the set, Γ_(x)¹,

[0230] of the cyclic frequencies γ.

[0231] The detection of the second-order cyclic frequencies of theobservations, α_(ε), for ε=−1 and ε=+1 by any detector of cyclicfrequencies and the constitution of the estimates,Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]),

[0232] of the sets respectively,Γ_(x)^([1, −1])  and  Γ_(x)^([1, 1]),

[0233] of the cyclic frequencies respectively of the first and secondmatrices of correlation of the observations.

[0234] The estimation of the matrix R_(Δx)(0) by {circumflex over(R)}_(Δx)(0)(K) defined by (33) and (32) for l=0 from a given number ofsamples K.

[0235] The detection of the number of sources P from the decompositionof {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All thenon-deterministic sources are detected).

[0236] The computation of the whitening matrix of the observations,{circumflex over (T)}, where {circumflex over (T)} ^(Δ) {circumflex over(Λ)}_(s) ^(−1/2) Û_(s) ^(\), with a size (P×N), where {circumflex over(Λ)}_(s) is the diagonal matrix (P×P) of the P greatest eigen values of{circumflex over (R)}_(Δx)(0)(K)−λmin I, λmin is the minimum eigen valueof {circumflex over (R)}_(Δx)(0)(K) and Û_(s) is the matrix of theassociated eigen vectors. We write z(t) ^(Δ) {circumflex over (T)} x(t).(The directional vectors of the non-deterministic sources areorthonormalized).

[0237] Identification Step

[0238] The choice of a triplet (α_(m), ε_(m), l_(mq))

[0239] The estimation of the matrix of second-order cyclic cumulants ofthe whitened observations,R_(Δ  z  ɛ_(m)  )^(α_(m))(l_(mq)T_(e)), by  R̂_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)  

[0240] defined by (31), (29), (32).

[0241] The computation of a unit matrix Û_((αm,εm))=[e₁. . . e_(p),]where(P_((αm, εm)≦P)| corresponds to the number of non-zero eigen values of)R̂_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)

[0242] and e_(k) (1≦k≦P′) are the eigen vectors ofR̂_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)

[0243] associated with the P_((αm, εm)) highest eigen values.

[0244] The choice of the set (α_(m), ζ_(m), l_(m1)T_(e), l_(m2)T_(e),l_(m3)T_(e)).

[0245] The reduction of dimension: v(t) ^(Δ Û) _((αm, εm)) ^(†)z(t) iswritten, with a dimension (P_((αm, εm))×1) and a computation is made ofthe estimate of the cyclic quadricovarianceQ̂_(v  ζ  m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)

[0246] of v(t) fromQ̂_(v  ζ  m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)

[0247] and from Û_((αm, ζm)).

[0248] The decomposition into eigen elements ofQ̂_(v  ζ  m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)

[0249] and the estimation of the P_((αm, εm)) eigen matrices M_(vi)(1≦i≦P_((αm, εm))) associated with the P_((αm, εm)) eigen values withhigher-value moduli.

[0250] The joint diagonalization of the P_((αm, εm)) eigen matricesM_(vi)M_(vi) ^(†)weighted by the associated eigen values and theobtaining of the matrix of the directional vectors of the doublywhitened non-deterministic sources associated with the set (α_(m),ζ_(m), l_(m1), l_(m2), l₃): Â_(nd) _((αm, ζm))′

[0251] Filter

[0252] The computation of an estimate of the matrix of the directionalvectors of the non-deterministic sources associated with the pair(α_(m), ζ_(m)): Â_(nd) _((αm, ζm)) =Û_(s){circumflex over (Λ)}_(s)^(1/2)Û_((αm, εm))Â_(nd) _((αm, εm)) ′

[0253] The concatenation of the matrices Â_(nd) _((αm, ζm)) for all thepairs (α_(m), ζ_(m)), (1≦m≦M). The matrix (N×P) Â_(nd) of thedirectional vectors of the non-deterministic sources is obtained.

[0254] The extraction of the non-deterministic sources by any spatialfiltering whatsoever of the observations constructed on the basis ofÂ_(nd).

[0255] Processing of the Deterministic Sources

[0256] The construction of the orthogonal projector on the spaceorthogonal to the columns of Â_(nd):Â_(nd) : Proj = I − Â_(nd)[Â_(nd)^(†)Â_(nd)]⁻¹Â_(nd)^(†)

[0257] The implementation of the algorithm JADE [4] from theobservations w(t) ^(Δ) Proj x(t) to identify the directional vectors ofthe deterministic sources and extract these sources.

[0258] Exemplary Simulation of the Method According to the Inventionwith the Separator F3′

[0259]FIGS. 6A and 6B respectively give a view, in a level graphexpressed in dB, for two non-centered cyclostationary sources with adirection (↓₁=50° and ↓₂=60°), of the spectrum level after separation ofthe sources (in using the method JADE bringing into play the set offirst-order cyclic frequencies and the two sets of second-order cyclicfrequencies) and of the channel formation, after separation of thesources by using the classic estimator (FIG. 6A) and by using theproposed estimator (FIG. 6B).

[0260] It can be seen that, with the empirical estimator, the separationworks badly because a source is localized at ↓=55° while the sourceshave angles of incidence of ↓₁=50° and ↓₂=60°. However, with theproposed estimator, the two sources are localized at 50° and 60° becausethe channel formation method on the first identified vector finds amaximum at 49.8° and, on the second vector, it finds a maximum at 60.1°.The signal-to-noise+interference ratios at output of filtering of thetwo sources are summarized in the following table: Source at θ₁ = 50°Source at θ₂ = 60° Empirical estimators SNIR₁ = 15.3 dB SNIR₂ = 7.35 dBR_(xa) and Q_(xa) Proposed estimators SNIR₁ = 22 dB SNIR₂ = 22 dB R_(x)and Q_(x)

[0261] In the following simulation, the proposed estimator is known asan exhaustive estimator. This example keeps the same signalconfiguration as earlier when the two sources nevertheless have asignal-to-noise ratio of 10 dB. The classic empirical estimator istherefore compared with the exhaustive estimator. The SNIR in dB of thefirst source is therefore plotted as a function of the number of samplesK.

[0262]FIG. 7 shows that the exhaustive estimator converges on theasymptote of the optimum interference canceling filter when K→+∞ andthat the classic estimator is biased in converging on an SNIR(signal-to-noise+interference ratio) of 5 dB at output of filtering.

[0263] Separators F4′

[0264] The separators of the family F4′ are fourth-order self-learningseparators implementing the following operations:

[0265] The detection of the first-order cyclic frequencies of theobservations γ by any detector whatsoever of cyclic frequencies and theconstitution of an estimate Γ̂_(x)¹,

[0266] of the set, Γ_(x)¹,

[0267] of the cyclic frequencies γ. The estimation of Γ̂_(x)¹,

[0268] can be done as in F1′.

[0269] The detection of the second-order cyclic frequencies of theobservations, α_(ε), for ε=−1 and ε=+1 by any detector whatsoever ofcyclic frequencies and the constitution of the estimates,Γ̂_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]),

[0270] of the sets respectively,Γ_(x)^([1, −1])  and  Γ̂_(x)^([1, 1]),

[0271] of the cyclic frequencies respectively of the first and secondmatrices of correlation of the observations.

[0272] The estimation of the matrix R_(Δx)(0) by {circumflex over(R)}_(Δx)(0)(K) defined by (53) and (52) for l=0 from a given number ofsamples K.

[0273] The detection of the number of sources P from the decompositionof {circumflex over (R)}_(Δx)(0)(K) into eigen elements. (All thenon-deterministic sources are detected).

[0274] The computation of the whitening matrix of the observations,{circumflex over (T)}, where {circumflex over (T)}^(Δ {circumflex over (Λ)}) _(s) ^(−1/2) Û_(s) ^(†), with a size (P×N),where {circumflex over (Λ)}_(s) is the diagonal matrix (P×P) of the Pgreatest eigen values of {circumflex over (R)}_(Δx)(0)(K)−λmin I, λminis the minimum eigen value of {circumflex over (R)}_(Δx)(0)(K) and Û_(s)is the matrix of the associated eigen vectors. We write z(t) ^(Δ){circumflex over (T)} x(t). (The directional vectors of thenon-deterministic sources are orthonormalized).

[0275] The choice of a triplet (α_(m), ε_(m), l_(mq))

[0276] The estimation of the matrix of second-order cyclic cumulants ofthe whitened observations,R_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e)), by  R̂_(Δ  z  ɛ_(m))^(α_(m))(l_(mq)T_(e))(K)

[0277] defined by (31), (29), (32).

[0278] The computation of a unit matrixÛ_((α  m, ɛ  m)) = [e_(1  )⋯  e_(p^(′))]  where  (P_((α  m  , ɛ  m)) ≤ P)

[0279] corresponds to the number of non-zero eigen valuesof  R̂_(Δ  z  ɛ)^(α_(m)),  _(m)(l_(mq)T_(e))(K)

[0280] and e_(k) (1≦k≦P_((αm, εm))) are the eigen vectors ofR̂_(Δ  z  ɛ)^(α_(m)),  _(m)(l_(mq)T_(e))(K)

[0281] associated with the P_((αm, εm))) higher eigen values.

[0282] The reduction of dimension: v(t) ^(Δ) Û_((αm, εm)) ^(†)z(t) iswritten with a dimension (P_((αm, εm))×1) and a computation is made ofthe estimate of the cyclic quadricovarianceQ̂_(v  ζ  m)^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)  of  v(t)

[0283] fromQ̂_(v  ζ  m  )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)  and  from  Û_((α  m, ζ  m)),

[0284] The choice of a set (α_(m), ζ_(m), l_(m1), l_(m2), l₃) (1≦m≦M)such that, for each of these pairs, at least one source possesses thefourth-order cyclic frequency α_(m) for a quadricovariance indexed byζ_(m) and (l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e)),

[0285] The set of these values is chosen, for example, so that there iscompatibility with the parameters of the second-order cumulants (α_(m),ε_(m), l_(mq))

[0286] For each set (α_(m), ζ_(m), l_(m1), l_(m2), l_(m3)) (1≦m≦M)

[0287] The estimation of the cyclic quadricovariance,Q̂_(v  ζ  m  )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e)),

[0288] of the vector v(t) . We obtainQ̂_(v  ζ  m  )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K).

[0289] The decomposition into eigen elements ofQ̂_(v  ζ  m  )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)

[0290] (K) and the estimation of the P_((αm, εm)) eigen matricesM_(vi)(1≦i≦P_((αm, εm))) associated with the P_((αm, εm)) eigen valueswith higher-value moduli.

[0291] The joint diagonalization of the P_((αm, εm)) eigen matricesM_(vi)M_(vi) ^(†)weighted by the associated eigen values and theobtaining of the matrix of the directional vectors of the doublywhitened non-deterministic sources A associated with the set (α_(m),ζ_(m), l_(m1), l_(m2), l_(m3)): Â_(nd) _((αm, ζm)) ′

[0292] The computation of an estimate of the matrix of the directionalvectors of the non-deterministic sources associated with the pair(α_(m), ζ_(m)): Â_(nd) _((αm, ζm)) =Û_(s){circumflex over (Λ)}_(s)^(1/2)Û_((αm, εm))Â_(nd) _((αm, εm)) ′

[0293] The concatenation of the matrices Â_(nd) _((αm, ζm)) for all thepairs (α_(m), ζ_(m)), (1≦m≦M). The matrix (N×P) Â_(nd) of thedirectional vectors of the non-deterministic sources is obtained.

[0294] The extraction of the non-deterministic sources by any spatialfiltering whatsoever of the observations constructed on the basis ofÂ_(nd).

[0295] The construction of the orthogonal projector on the spaceorthogonal to the columns of Â_(nd): Proj=I−Â_(nd)[Â_(nd)^(†)Â_(nd)]⁻¹Â_(nd) ^(†)

[0296] The implementation of the algorithm JADE [4] from theobservations w(t) ^(Δ) Proj x(t) to identify the directional vectors ofthe deterministic sources and extract them.

[0297] The implementation is summarized in FIG. 8 for M=1 (only one set(α_(m), ζ_(m), l_(m1), l_(m2), l_(m3)) is used).

[0298] The steps of the method according to the invention described hereabove are applied especially in the above-mentioned separationtechniques, namely the SOBI, cyclic SOBI, JADE and cyclic JADEtechniques.

[0299] Without departing from the scope of the invention, the steps ofthe method described here above are used, for example, to carry out theangular localization or goniometry of signals received at the level of areceiver.

[0300] For classic angular localization, the method uses for example:

[0301] MUSIC type methods described in the reference [10] of R. OSchmidt with the matrix of covariance: {circumflex over (R)}_(Δx)(0)(K)

[0302] Cyclic goniometry type methods given in the reference [11] by W.A. Gardner with the matrix of cyclic covariance: R_(Δxε), ^(α) ^(_(m))_(m)(l_(mq)T_(e))

[0303] Fourth-order goniometry methods described in the reference [12]by P. Chevalier, A. Ferreol, J P. Denis with the quadricovariance:{circumflex over (Q)}_(xζ), ⁰ _(m)(0, 0, 0)(K)

[0304] Methods of fourth-order cyclic goniometry with the cyclicquadricovariance:Q̂_(x  ζ  m  )^(α_(m))(l_(m1)T_(e), l_(m2)T_(e), l_(m3)T_(e))(K)

Bibliography

[0305] [1] C. JUTTEN, J. HERAULT, <<Blind separation of sources, Part I:An adaptive algorithm based on neuromimetic architecture>>, SignalProcessing, Elsevier, Vol 24, pp 1-10, 1991.

[0306] [2] P. COMON, P. CHEVALIER, <<Blind source separation: Models,Concepts, Algorithms and Performance>>, Chapter 5 of the bookUnsupervised adaptive filtering—Tome 1—Blind Source Separation, pp.191-235, Dir. S. Haykins, Wiley, 445 p, 2000.

[0307] [3] A. BELOUCHRANI, K. ABED-MERAIM, J. F. CARDOSO, E. MOULINES,<<A blind source separation technique using second-order statistics>>,IEEE Tran. Signal Processing, Vol. 45, N^(o). 2, pp 434-444, February1997.

[0308] [4] J. F. CARDOSO, A. SOULOUMIAC, <<Blind beamforming fornon-Gaussian signals>>, IEEE Proceedings-F, Vol. 140, N^(o). 6, pp362-370, December 1993.

[0309] [5] A. FERREOL, P. CHEVALIER, <<On the behavior of current secondand higher order blind source separation methods for cyclostationarysources>>, IEEE Trans. Signal Processing, Vol 48, N^(o). 6, pp.1712-1725, June 2000.

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[0311] [7] P. CHEVALIER, <<Optimal separation of independent narrow-bandsources—Concept and Performance>>, Signal Processing, Elsevier, Specialissue on Blind Source Separation and Multichannel Deconvolution, Vol 73,N^(o). 1-2, pp. 27-47, February 1999.

[0312] [8] A. FERREOL, P. CHEVALIER, <<Higher Order Blind SourceSeparation using the Cyclostationarity Property of the Signals>>,ICASSP, Munich (Germany), pp 4061-4064, April 1997.

[0313] [9] SV SCHELL and W. GARDNER, “Detection of the number ofcyclostationnary signals in unknown interference and noise”, Proc ofAsilonan conf on signal, systems and computers 5-7 November 90.

[0314] [10] R. O Schmidt, A signal subspace approach to multipleemitters location and spectral estimation, PhD Thesis, StanfordUniversity, CA, November 1981.

[0315] [11] W. A. Gardner, “Simplification of MUSIC and ESPRIT byexploitation cyclostationarity”, Proc. IEEE, Vol. 76 No. 7, July 1988.

[0316] [12] P. Chevalier, A. Ferreol, J P. Denis, “New geometricalresults about 4-th order direction finding methods performances”,EUSIPCO, Trieste, pp. 923-926, 1996.

What is claimed is:
 1. An antenna processing method for centered orpotentially non-centered cyclostationary signals, comprising at leastone step in which one or more nth order estimators are obtained fromr-order statistics, with r=1 to n−1, and for one or more values of r, astep for the correction of the estimator by means of r-order detectedcyclic frequencies.
 2. A method according to claim I comprising a stepfor the separation of the emitter sources of the signals received byusing the estimator or estimators.
 3. A method according to claim 2wherein the estimator is a second-order estimator.
 4. A method accordingto claim 2 wherein the estimator is a fourth-order estimator.
 5. Amethod according to one of the claims 2 to 4 wherein the cyclicfrequencies are detected first-order or second-order frequencies.
 6. Ause of the method according to one of the claims 1 to 5 to separatepotentially non-centered cyclostationary sources in SOBI, or cyclicSOBI, JADE, or cyclic JADE separation techniques.
 7. A use of the methodaccording to claim 1 to carry out the angular localization of thesignals received.